:: PDIFF_7 semantic presentation

begin

registration
let n be ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ;
let p, q be ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of ( ( ) ( non empty ) set ) ) ;
let f, g be ( ( Relation-like Function-like FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) ;
identify ;
end;

registration
let r, s be ( ( real ) ( V11() real ext-real ) number ) ;
let n be ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ;
let p be ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of ( ( ) ( non empty ) set ) ) ;
let f be ( ( Relation-like Function-like FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) ;
identify ;
end;

registration
let n be ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ;
let p be ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of ( ( ) ( non empty ) set ) ) ;
let f be ( ( Relation-like Function-like FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) ;
identify ;
end;

registration
let n be ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ;
let p, q be ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite n : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of ( ( ) ( non empty ) set ) ) ;
let f, g be ( ( Relation-like Function-like FinSequence-like real-valued ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence) ;
identify ;
end;

theorem :: PDIFF_7:1
for i, j being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(0* j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) | i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = 0* i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:2
for i, j being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(0* j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) | (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = 0* (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL (b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:3
for j, i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds (0* j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /^ i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = 0* (j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL (b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:4
for i, j being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( ( i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) implies (0* j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) | (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = 0* (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL (b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) & (0* j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /^ i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = 0* (j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL (b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) ;

theorem :: PDIFF_7:5
for i, j being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for xi being ( ( ) ( ) Element of ( ( ) ( non empty V2() ) set ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
||.((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0. (REAL-NS j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) : ( ( ) ( V52( REAL-NS b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . xi : ( ( ) ( ) Element of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b2 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = ||.xi : ( ( ) ( ) Element of ( ( ) ( non empty V2() ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ;

theorem :: PDIFF_7:6
for m, i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for x being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )
for r being ( ( ) ( V11() real ext-real ) Real) holds
( ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0* m : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (r : ( ( ) ( V11() real ext-real ) Real) - ((proj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,m : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) & x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0* m : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (((proj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,m : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) - r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) ;

theorem :: PDIFF_7:7
for m, i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for p being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) holds
( ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . p : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) - x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) = (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0. (REAL-NS m : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) : ( ( ) ( V52( REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (p : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) - ((Proj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,m : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) - ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . p : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) = (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0. (REAL-NS m : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) : ( ( ) ( V52( REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (((Proj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,m : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) - p : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: PDIFF_7:8
for m, n being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for Z being ( ( ) ( ) Subset of ) st Z : ( ( ) ( ) Subset of ) is open & 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_on Z : ( ( ) ( ) Subset of ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) iff ( Z : ( ( ) ( ) Subset of ) c= dom f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) : ( ( ) ( ) Element of K6( the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in Z : ( ( ) ( ) Subset of ) holds
f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ) ;

theorem :: PDIFF_7:9
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for x, y being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
Replace ((0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) + y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = (Replace ((0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) + (Replace ((0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_7:10
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for x, a being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
Replace ((0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) * x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) * (Replace ((0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ;

theorem :: PDIFF_7:11
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for x being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) <> 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below V263() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
Replace ((0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) <> 0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:12
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for x, y being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) )
for z being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = (proj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) holds
( (Replace (z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) - z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = Replace ((0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) - y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) & z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - (Replace (z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of K6(K7(NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = Replace ((0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) - x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:13
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for x, y being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) + y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) + ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:14
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for x, y being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0. (REAL-NS m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) : ( ( ) ( V52( REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) + y : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) = ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0. (REAL-NS m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) : ( ( ) ( V52( REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) + ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0. (REAL-NS m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) : ( ( ) ( V52( REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . y : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ;

theorem :: PDIFF_7:15
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for x, a being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) * x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) * ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:16
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for a being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0. (REAL-NS m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) : ( ( ) ( V52( REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) * x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) * ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0. (REAL-NS m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) : ( ( ) ( V52( REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ;

theorem :: PDIFF_7:17
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for x being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) <> 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below V263() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) <> 0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:18
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) <> 0. (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( V52( REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) holds
(reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0. (REAL-NS m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) : ( ( ) ( V52( REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) <> 0. (REAL-NS m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( V52( REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ;

theorem :: PDIFF_7:19
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for x, y being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) )
for z being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = (proj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) holds
( ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) - y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) & z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) - x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) ;

theorem :: PDIFF_7:20
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for x, y being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for z being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & y : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = (Proj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . z : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) holds
( ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) - z : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) = (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0. (REAL-NS m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) : ( ( ) ( V52( REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) - y : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) & z : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) - ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) = (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0. (REAL-NS m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) : ( ( ) ( V52( REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . (y : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) - x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) ) ;

theorem :: PDIFF_7:21
errorfrm ;

theorem :: PDIFF_7:22
for n, m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for g being ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,)
for y being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st g : ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) & 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( g : ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_partial_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & partdiff (g : ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = ((diff (g : ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7((REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0. (REAL-NS m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) : ( ( ) ( V52( REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) ) ) Element of the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . <*1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( non empty V2() Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like one-to-one constant finite 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) set ) ) ;

definition
let n be ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL n : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let x be ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite n : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;
pred f is_differentiable_in x means :: PDIFF_7:def 1
 <>* f : ( ( ) ( ) Element of n : ( ( ) ( ) NORMSTR ) ) : ( ( Function-like ) ( Relation-like REAL n : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7((REAL n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,(REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( Function-like quasi_total ) ( Relation-like K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
let n be ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL n : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let x be ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite n : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;
func diff (f,x) -> ( ( Function-like quasi_total ) ( non empty Relation-like REAL n : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL n : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) , REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) equals :: PDIFF_7:def 2
(proj (1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) * (diff ((<>* f : ( ( ) ( ) Element of n : ( ( ) ( ) NORMSTR ) ) ) : ( ( Function-like ) ( Relation-like REAL n : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7((REAL n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,(REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( Function-like quasi_total ) ( Relation-like K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(n : ( ( ) ( ) NORMSTR ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( Function-like quasi_total ) ( Relation-like REAL n : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7((REAL n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,(REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL n : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL n : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;
end;

theorem :: PDIFF_7:23
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for h being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for y being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st h : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) & 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( h : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & partdiff (h : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = diff ((h : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) * (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,((proj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) & partdiff (h : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = (diff (h : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) , REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) . ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(0* m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:24
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for v, w, u being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st dom v : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of K6(NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) = dom w : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of K6(NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & u : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = v : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) + w : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
Sum u : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (Sum v : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) + (Sum w : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:25
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( ) ( V11() real ext-real ) Real)
for w, u being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st u : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = r : ( ( ) ( V11() real ext-real ) Real) (#) w : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( Function-like ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
Sum u : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = r : ( ( ) ( V11() real ext-real ) Real) * (Sum w : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:26
for n being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for h, g being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL n : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st len h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) = (len g : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for i being ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) st i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) in dom g : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of K6(NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
g : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /. i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /. i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - (h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /. (i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) + 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) holds
(h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /. 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - (h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /. (len h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = Sum g : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:27
for n being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for h, g, j being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL n : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st len h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) = len j : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & len g : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) = len j : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for i being ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) st i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) in dom j : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of K6(NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
j : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /. i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /. i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - (g : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /. i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) holds
Sum j : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (Sum h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - (Sum g : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:28
for m, n being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,)
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ex h being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ex g being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL n : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st
( len h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) = m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & len g : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) = m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for i being ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) st i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) in dom h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of K6(NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /. i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) | ((m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ^ (0* (i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) -' 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b8 : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) -' 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL (b8 : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) -' 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) & ( for i being ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) st i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) in dom g : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of K6(NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
g : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /. i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. (x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) + (h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /. i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. (x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) + (h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /. (i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) + 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) & ( for i being ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat)
for hi being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) in dom h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of K6(NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /. i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = hi : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) holds
|.hi : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) <= |.y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) & (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. (x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) + y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = Sum g : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) ;

theorem :: PDIFF_7:29
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ex f0 being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) = <>* f0 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,(REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_7:30
for m, n being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,)
for f0 being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )
for x0 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) in dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Element of K6((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) & x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = x0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) = f0 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds
f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = f0 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) /. x0 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ;

definition
let m be ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let X be ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) ;
attr X is open means :: PDIFF_7:def 3
 ex X0 being ( ( ) ( ) Subset of ) st
( X0 : ( ( ) ( ) Subset of ) = X : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) & X0 : ( ( ) ( ) Subset of ) is open );
end;

theorem :: PDIFF_7:31
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) holds
( X : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) is open iff for x being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) holds
ex r being ( ( ) ( V11() real ext-real ) Real) st
( r : ( ( ) ( V11() real ext-real ) Real) > 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below V263() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & { y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) where y is ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : |.(y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) } c= X : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) ) ) ;

definition
let m, n be ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let i be ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL n : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
pred f is_partial_differentiable_on X,i means :: PDIFF_7:def 4
( X : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) NORMSTR ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) NORMSTR ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like functional FinSequence-membered ) Element of K6((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite m : ( ( ) ( ) NORMSTR ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) in X : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) NORMSTR ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) NORMSTR ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) | X : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) NORMSTR ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) NORMSTR ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL n : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like ) Element of K6(K7((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,(REAL n : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) );
end;

theorem :: PDIFF_7:32
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( ) set )
for m, n being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b3 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b4 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like REAL b3 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b4 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( ) set ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
X : ( ( ) ( ) set ) is ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) ;

theorem :: PDIFF_7:33
for m, n being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for Z being ( ( ) ( ) set ) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) = g : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_partial_differentiable_on Z : ( ( ) ( ) set ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) iff g : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_on Z : ( ( ) ( ) set ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: PDIFF_7:34
for m, n being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,)
for Z being ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) st Z : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) is open & 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_partial_differentiable_on Z : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) iff ( Z : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Element of K6((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) in Z : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ) ;

definition
let m, n be ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let i be ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL n : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
assume f : ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL n : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( ) set ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ;
func f `partial| (X,i) -> ( ( Function-like ) ( Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL n : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) means :: PDIFF_7:def 5
( dom it : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional FinSequence-membered ) Element of K6((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) = X : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) NORMSTR ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) NORMSTR ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite m : ( ( ) ( ) NORMSTR ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) in X : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) NORMSTR ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) NORMSTR ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) holds
it : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) /. x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) Element of REAL n : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = partdiff (f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) );
end;

definition
let m, n be ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL n : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
let x0 be ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;
pred f is_continuous_in x0 means :: PDIFF_7:def 6
 ex y0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ex g being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( ) ( ) NORMSTR ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS n : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) st
( x0 : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = y0 : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) & f : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) & g : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in y0 : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) );
end;

theorem :: PDIFF_7:35
for m, n being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )
for y being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) = g : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) & x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_continuous_in x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) iff g : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_in y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: PDIFF_7:36
for m, n being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,)
for x0 being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_continuous_in x0 : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) iff ( x0 : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) in dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Element of K6((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) & ( for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below V263() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below V263() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st x1 : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) in dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Element of K6((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) & |.(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - x0 : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
|.((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. x1 : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. x0 : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

definition
let m, n be ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL n : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
pred f is_continuous_on X means :: PDIFF_7:def 7
( X : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like functional FinSequence-membered ) Element of K6((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite m : ( ( ) ( ) NORMSTR ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st x0 : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) in X : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) NORMSTR ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) | X : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL m : ( ( ) ( ) NORMSTR ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL n : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like ) Element of K6(K7((REAL m : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,(REAL n : ( ( ) ( ) Element of m : ( ( ) ( ) NORMSTR ) ) ) : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_continuous_in x0 : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) );
end;

theorem :: PDIFF_7:37
for m, n being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,)
for X being ( ( ) ( ) set ) st f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) = g : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff g : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V185() ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_7:38
for m, n being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,)
for X being ( ( ) ( ) set ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( ) set ) iff ( X : ( ( ) ( ) set ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Element of K6((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )
for r being ( ( ) ( V11() real ext-real ) Real) st x0 : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) in X : ( ( ) ( ) set ) & 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below V263() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) holds
ex s being ( ( ) ( V11() real ext-real ) Real) st
( 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below V263() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) < s : ( ( ) ( V11() real ext-real ) Real) & ( for x1 being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st x1 : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) in X : ( ( ) ( ) set ) & |.(x1 : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - x0 : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) < s : ( ( ) ( V11() real ext-real ) Real) holds
|.((f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. x1 : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) /. x0 : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) ) ) ) ) ) ;

theorem :: PDIFF_7:39
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for xi being ( ( ) ( V11() real ext-real ) Real) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . xi : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) holds
(proj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = xi : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: PDIFF_7:40
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for xi being ( ( ) ( V11() real ext-real ) Real) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . xi : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) holds
reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_7:41
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for x, y being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for xi being ( ( ) ( V11() real ext-real ) Real) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . xi : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) & g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) * (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) holds
diff (g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,xi : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = partdiff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ;

theorem :: PDIFF_7:42
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for p, q being ( ( ) ( V11() real ext-real ) Real)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( V11() real ext-real ) Real) < q : ( ( ) ( V11() real ext-real ) Real) & ( for h being ( ( ) ( V11() real ext-real ) Real) st h : ( ( ) ( V11() real ext-real ) Real) in [.p : ( ( ) ( V11() real ext-real ) Real) ,q : ( ( ) ( V11() real ext-real ) Real) .] : ( ( ) ( complex-membered ext-real-membered real-membered closed V263() ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) holds
(reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . h : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) in dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Element of K6((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) ) & ( for h being ( ( ) ( V11() real ext-real ) Real) st h : ( ( ) ( V11() real ext-real ) Real) in [.p : ( ( ) ( V11() real ext-real ) Real) ,q : ( ( ) ( V11() real ext-real ) Real) .] : ( ( ) ( complex-membered ext-real-membered real-membered closed V263() ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . h : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
ex r being ( ( ) ( V11() real ext-real ) Real) ex y being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st
( r : ( ( ) ( V11() real ext-real ) Real) in ].p : ( ( ) ( V11() real ext-real ) Real) ,q : ( ( ) ( V11() real ext-real ) Real) .[ : ( ( ) ( complex-membered ext-real-membered real-membered open non left_end non right_end V263() ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) & y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) & (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . q : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) - (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . p : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = (q : ( ( ) ( V11() real ext-real ) Real) - p : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) * (partdiff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:43
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for p, q being ( ( ) ( V11() real ext-real ) Real)
for x being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( V11() real ext-real ) Real) <= q : ( ( ) ( V11() real ext-real ) Real) & ( for h being ( ( ) ( V11() real ext-real ) Real) st h : ( ( ) ( V11() real ext-real ) Real) in [.p : ( ( ) ( V11() real ext-real ) Real) ,q : ( ( ) ( V11() real ext-real ) Real) .] : ( ( ) ( complex-membered ext-real-membered real-membered closed V263() ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) holds
(reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . h : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) in dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Element of K6((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) ) & ( for h being ( ( ) ( V11() real ext-real ) Real) st h : ( ( ) ( V11() real ext-real ) Real) in [.p : ( ( ) ( V11() real ext-real ) Real) ,q : ( ( ) ( V11() real ext-real ) Real) .] : ( ( ) ( complex-membered ext-real-membered real-membered closed V263() ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . h : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
ex r being ( ( ) ( V11() real ext-real ) Real) ex y being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st
( r : ( ( ) ( V11() real ext-real ) Real) in [.p : ( ( ) ( V11() real ext-real ) Real) ,q : ( ( ) ( V11() real ext-real ) Real) .] : ( ( ) ( complex-membered ext-real-membered real-membered closed V263() ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) & y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) & (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . q : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) - (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. ((reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . p : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = (q : ( ( ) ( V11() real ext-real ) Real) - p : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) * (partdiff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:44
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for x, y, z, w being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for d, p, q, r being ( ( ) ( V11() real ext-real ) Real) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & |.(y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) < d : ( ( ) ( V11() real ext-real ) Real) & |.(z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) < d : ( ( ) ( V11() real ext-real ) Real) & p : ( ( ) ( V11() real ext-real ) Real) = (proj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) & z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . q : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) & r : ( ( ) ( V11() real ext-real ) Real) in [.p : ( ( ) ( V11() real ext-real ) Real) ,q : ( ( ) ( V11() real ext-real ) Real) .] : ( ( ) ( complex-membered ext-real-membered real-membered closed V263() ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) & w : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) holds
|.(w : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) < d : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: PDIFF_7:45
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for X being ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) )
for x, y, z being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )
for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for d, p, q being ( ( ) ( V11() real ext-real ) Real) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & X : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) is open & x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) & |.(y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) < d : ( ( ) ( V11() real ext-real ) Real) & |.(z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) < d : ( ( ) ( V11() real ext-real ) Real) & X : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Element of K6((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) & 0 : ( ( ) ( empty ordinal natural V11() real ext-real non positive non negative Function-like functional FinSequence-membered V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below V263() complex-functions-membered ext-real-functions-membered real-functions-membered rational-functions-membered integer-functions-membered natural-functions-membered ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) < d : ( ( ) ( V11() real ext-real ) Real) & ( for z being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st |.(z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) < d : ( ( ) ( V11() real ext-real ) Real) holds
z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) ) & z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (reproj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . p : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) & q : ( ( ) ( V11() real ext-real ) Real) = (proj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) holds
ex w being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st
( |.(w : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) - x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) .| : ( ( ) ( V11() real ext-real non negative ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) < d : ( ( ) ( V11() real ext-real ) Real) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in w : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. z : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) - (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) /. y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) = (p : ( ( ) ( V11() real ext-real ) Real) - q : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) * (partdiff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,w : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:46
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for h being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )
for y, x being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) )
for j being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st len h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) = m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for i being ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) st i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) in dom h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of K6(NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /. i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) | ((m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ^ (0* (i : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) -' 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b6 : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) -' 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL (b6 : ( ( natural ) ( ordinal natural V11() real ext-real non negative ) Nat) -' 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) holds
x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) + (h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /. j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = (reproj (((m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,(x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) + (h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) /. (j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ,(REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . ((proj (((m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -' j : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) + y : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ;

theorem :: PDIFF_7:47
for m being ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,)
for X being ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) )
for x being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st X : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) is open & x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) in X : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) & ( for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) & i : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) <= m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) & f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) `partial| (X : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) ,i : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_continuous_on X : ( ( ) ( functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) Subset of ( ( ) ( ) set ) ) ) ) holds
( f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) is_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) & ( for h being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ex w being ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) st
( dom w : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of K6(NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) = Seg m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of K6(NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) & ( for i being ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) in Seg m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end right_end bounded_below bounded_above real-bounded ) Element of K6(NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
w : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) . i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) set ) = ((proj (i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ,m : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) * (partdiff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ,i : ( ( ) ( ordinal natural V11() real ext-real non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) & (diff (f : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like complex-functions-valued ext-real-functions-valued real-functions-valued ) PartFunc of ,) ,x : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like total quasi_total complex-functions-valued ext-real-functions-valued real-functions-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ,(REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . h : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) = Sum w : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) -valued Function-like finite FinSequence-like FinSubsequence-like complex-functions-valued ext-real-functions-valued real-functions-valued ) FinSequence of REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) -valued Function-like finite 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued ) Element of REAL 1 : ( ( ) ( non empty ordinal natural V11() real ext-real positive non negative V163() V198() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered left_end bounded_below ) Element of NAT : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V205() bounded_below ) Element of K6(REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty functional FinSequence-membered complex-functions-membered ext-real-functions-membered real-functions-membered ) FinSequenceSet of REAL : ( ( ) ( non empty non finite complex-membered ext-real-membered real-membered V205() non bounded_below non bounded_above V263() ) set ) ) ) ) ) ) ;

theorem :: PDIFF_7:48
errorfrm ;

theorem :: PDIFF_7:49
errorfrm ;