:: SUPINF_2 semantic presentation

begin

notation
synonym 0. for 0 ;
end;

definition
:: original: 0.
redefine func 0. -> ( ( ) ( ext-real ) R_eal) ;
let x be ( ( ) ( ext-real ) R_eal) ;
:: original: -
redefine func - x -> ( ( ) ( ext-real ) R_eal) ;
let y be ( ( ) ( ext-real ) R_eal) ;
:: original: +
redefine func x + y -> ( ( ) ( ext-real ) R_eal) ;
:: original: -
redefine func x - y -> ( ( ) ( ext-real ) R_eal) ;
end;

theorem :: SUPINF_2:1
for x, y being ( ( ) ( ext-real ) R_eal)
for a, b being ( ( ) ( V35() V36() ext-real ) Real) st x : ( ( ) ( ext-real ) R_eal) = a : ( ( ) ( V35() V36() ext-real ) Real) & y : ( ( ) ( ext-real ) R_eal) = b : ( ( ) ( V35() V36() ext-real ) Real) holds
x : ( ( ) ( ext-real ) R_eal) + y : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) = a : ( ( ) ( V35() V36() ext-real ) Real) + b : ( ( ) ( V35() V36() ext-real ) Real) : ( ( ) ( V35() V36() ext-real ) Element of REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) ;

theorem :: SUPINF_2:2
for x being ( ( ) ( ext-real ) R_eal)
for a being ( ( ) ( V35() V36() ext-real ) Real) st x : ( ( ) ( ext-real ) R_eal) = a : ( ( ) ( V35() V36() ext-real ) Real) holds
- x : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) = - a : ( ( ) ( V35() V36() ext-real ) Real) : ( ( ) ( V35() V36() ext-real ) Element of REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) ;

theorem :: SUPINF_2:3
for x, y being ( ( ) ( ext-real ) R_eal)
for a, b being ( ( ) ( V35() V36() ext-real ) Real) st x : ( ( ) ( ext-real ) R_eal) = a : ( ( ) ( V35() V36() ext-real ) Real) & y : ( ( ) ( ext-real ) R_eal) = b : ( ( ) ( V35() V36() ext-real ) Real) holds
x : ( ( ) ( ext-real ) R_eal) - y : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) = a : ( ( ) ( V35() V36() ext-real ) Real) - b : ( ( ) ( V35() V36() ext-real ) Real) : ( ( ) ( V35() V36() ext-real ) Element of REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) ;

notation
let X, Y be ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ;
synonym X + Y for X ++ Y;
end;

definition
let X, Y be ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ;
:: original: +
redefine func X + Y -> ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ;
end;

notation
let X be ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ;
synonym - X for -- X;
end;

definition
let X be ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ;
:: original: -
redefine func - X -> ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ;
end;

theorem :: SUPINF_2:4
canceled;

theorem :: SUPINF_2:5
for X being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for z being ( ( ) ( ext-real ) R_eal) holds
( z : ( ( ) ( ext-real ) R_eal) in X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) iff - z : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) in - X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ;

theorem :: SUPINF_2:6
for X, Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) holds
( X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) c= Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) iff - X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) c= - Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ;

theorem :: SUPINF_2:7
for z being ( ( ) ( ext-real ) R_eal) holds
( z : ( ( ) ( ext-real ) R_eal) in REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) iff - z : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) in REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) ;

definition
let X be ( ( ext-real-membered ) ( ext-real-membered ) set ) ;
:: original: inf
redefine func inf X -> ( ( ) ( ext-real ) R_eal) ;
:: original: sup
redefine func sup X -> ( ( ) ( ext-real ) R_eal) ;
end;

theorem :: SUPINF_2:8
for X, Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) holds sup (X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) R_eal) <= (sup X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) + (sup Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) ;

theorem :: SUPINF_2:9
for X, Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) holds (inf X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) + (inf Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) <= inf (X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) R_eal) ;

theorem :: SUPINF_2:10
for X, Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) st X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) is bounded_above & Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) is bounded_above holds
sup (X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) R_eal) <= (sup X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) + (sup Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) ;

theorem :: SUPINF_2:11
for X, Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) st X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) is bounded_below & Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) is bounded_below holds
(inf X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) + (inf Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) <= inf (X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) R_eal) ;

theorem :: SUPINF_2:12
for X being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for a being ( ( ) ( ext-real ) R_eal) holds
( a : ( ( ) ( ext-real ) R_eal) is ( ( ) ( ext-real ) UpperBound of X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) iff - a : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) is ( ( ) ( ext-real ) LowerBound of - X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ;

theorem :: SUPINF_2:13
for X being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for a being ( ( ) ( ext-real ) R_eal) holds
( a : ( ( ) ( ext-real ) R_eal) is ( ( ) ( ext-real ) LowerBound of X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) iff - a : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) is ( ( ) ( ext-real ) UpperBound of - X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ;

theorem :: SUPINF_2:14
for X being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) holds inf (- X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) R_eal) = - (sup X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) ;

theorem :: SUPINF_2:15
for X being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) holds sup (- X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) R_eal) = - (inf X : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let Y be ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ;
let F be ( ( Function-like V18(X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ;
:: original: rng
redefine func rng F -> ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ;
end;

definition
let D be ( ( ext-real-membered ) ( ext-real-membered ) set ) ;
let X be ( ( ) ( ) set ) ;
let Y be ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ;
let F be ( ( Function-like ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like ext-real-valued ) PartFunc of ,) ;
func sup F -> ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) equals :: SUPINF_2:def 1
 sup (rng F : ( ( Function-like ) ( Relation-like D : ( ( complex-membered ) ( complex-membered ) set ) -defined X : ( ( V35() ) ( V35() ) set ) -valued Function-like ) Element of K19(K20(D : ( ( complex-membered ) ( complex-membered ) set ) ,X : ( ( V35() ) ( V35() ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K19(Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) R_eal) ;
func inf F -> ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) equals :: SUPINF_2:def 2
 inf (rng F : ( ( Function-like ) ( Relation-like D : ( ( complex-membered ) ( complex-membered ) set ) -defined X : ( ( V35() ) ( V35() ) set ) -valued Function-like ) Element of K19(K20(D : ( ( complex-membered ) ( complex-membered ) set ) ,X : ( ( V35() ) ( V35() ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K19(Y : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) R_eal) ;
end;

definition
let F be ( ( Relation-like Function-like ext-real-valued ) ( Relation-like Function-like ext-real-valued ) Function) ;
let x be ( ( ) ( ) set ) ;
:: original: .
redefine func F . x -> ( ( ) ( ext-real ) R_eal) ;
end;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let Y, Z be ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ;
let F be ( ( Function-like V18(X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ;
let G be ( ( Function-like V18(X : ( ( non empty ) ( non empty ) set ) ,Z : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined Z : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(X : ( ( non empty ) ( non empty ) set ) ,Z : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Z : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ;
func F + G -> ( ( Function-like V18(X : ( ( complex-membered ) ( complex-membered ) set ) ,Y : ( ( V35() ) ( V35() ) set ) + Z : ( ( ) ( ) set ) : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like X : ( ( complex-membered ) ( complex-membered ) set ) -defined Y : ( ( V35() ) ( V35() ) set ) + Z : ( ( ) ( ) set ) : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(X : ( ( complex-membered ) ( complex-membered ) set ) ,Y : ( ( V35() ) ( V35() ) set ) + Z : ( ( ) ( ) set ) : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( complex-membered ) ( complex-membered ) set ) ,Y : ( ( V35() ) ( V35() ) set ) + Z : ( ( ) ( ) set ) : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) means :: SUPINF_2:def 3
 for x being ( ( ) ( ) Element of X : ( ( complex-membered ) ( complex-membered ) set ) ) holds it : ( ( Function-like ) ( Relation-like Z : ( ( ) ( ) set ) -defined F : ( ( Function-like ) ( Relation-like X : ( ( complex-membered ) ( complex-membered ) set ) -defined Y : ( ( V35() ) ( V35() ) set ) -valued Function-like ) Element of K19(K20(X : ( ( complex-membered ) ( complex-membered ) set ) ,Y : ( ( V35() ) ( V35() ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like ) Element of K19(K20(Z : ( ( ) ( ) set ) ,F : ( ( Function-like ) ( Relation-like X : ( ( complex-membered ) ( complex-membered ) set ) -defined Y : ( ( V35() ) ( V35() ) set ) -valued Function-like ) Element of K19(K20(X : ( ( complex-membered ) ( complex-membered ) set ) ,Y : ( ( V35() ) ( V35() ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) R_eal) = (F : ( ( Function-like ) ( Relation-like X : ( ( complex-membered ) ( complex-membered ) set ) -defined Y : ( ( V35() ) ( V35() ) set ) -valued Function-like ) Element of K19(K20(X : ( ( complex-membered ) ( complex-membered ) set ) ,Y : ( ( V35() ) ( V35() ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) R_eal) + (G : ( ( ) ( V35() ) Element of X : ( ( complex-membered ) ( complex-membered ) set ) ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) ;
end;

theorem :: SUPINF_2:16
for X being ( ( non empty ) ( non empty ) set )
for Y, Z being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) )
for G being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Z : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) holds rng (F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) + G : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) c= (rng F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + (rng G : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ;

theorem :: SUPINF_2:17
for X being ( ( non empty ) ( non empty ) set )
for Y, Z being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) )
for G being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Z : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) holds sup (F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) + G : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) <= (sup F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) + (sup G : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( ) ( ext-real ) R_eal) ;

theorem :: SUPINF_2:18
for X being ( ( non empty ) ( non empty ) set )
for Y, Z being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) )
for G being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Z : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) holds (inf F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) + (inf G : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( ) ( ext-real ) R_eal) <= inf (F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) + G : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ;

definition
let X be ( ( non empty ) ( non empty ) set ) ;
let Y be ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ;
let F be ( ( Function-like V18(X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like X : ( ( non empty ) ( non empty ) set ) -defined Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ;
func - F -> ( ( Function-like V18(X : ( ( complex-membered ) ( complex-membered ) set ) , - Y : ( ( V35() ) ( V35() ) set ) : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like X : ( ( complex-membered ) ( complex-membered ) set ) -defined - Y : ( ( V35() ) ( V35() ) set ) : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(X : ( ( complex-membered ) ( complex-membered ) set ) , - Y : ( ( V35() ) ( V35() ) set ) : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( complex-membered ) ( complex-membered ) set ) , - Y : ( ( V35() ) ( V35() ) set ) : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) means :: SUPINF_2:def 4
 for x being ( ( ) ( ) Element of X : ( ( complex-membered ) ( complex-membered ) set ) ) holds it : ( ( Function-like ) ( Relation-like X : ( ( complex-membered ) ( complex-membered ) set ) -defined Y : ( ( V35() ) ( V35() ) set ) -valued Function-like ) Element of K19(K20(X : ( ( complex-membered ) ( complex-membered ) set ) ,Y : ( ( V35() ) ( V35() ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) R_eal) = - (F : ( ( ) ( ) set ) . x : ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) ;
end;

theorem :: SUPINF_2:19
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) holds rng (- F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) = - (rng F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ;

theorem :: SUPINF_2:20
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) holds
( inf (- F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) = - (sup F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( ) ( ext-real ) R_eal) & sup (- F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) = - (inf F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( ) ( ext-real ) R_eal) ) ;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ;
let F be ( ( Function-like V18(X : ( ( ) ( ) set ) ,Y : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like V18(X : ( ( ) ( ) set ) ,Y : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( ) ( ) set ) ,Y : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ;
attr F is bounded_above means :: SUPINF_2:def 5
 sup F : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) < +infty : ( ( ) ( V36() ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ;
attr F is bounded_below means :: SUPINF_2:def 6
 -infty : ( ( ) ( V36() ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) < inf F : ( ( ) ( ) set ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ;
end;

definition
let X be ( ( ) ( ) set ) ;
let Y be ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ;
let F be ( ( Function-like V18(X : ( ( ) ( ) set ) ,Y : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like X : ( ( ) ( ) set ) -defined Y : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like V18(X : ( ( ) ( ) set ) ,Y : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( ) ( ) set ) ,Y : ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ;
attr F is bounded means :: SUPINF_2:def 7
( F : ( ( ) ( ) set ) is bounded_above & F : ( ( ) ( ) set ) is bounded_below );
end;

registration
let X be ( ( ) ( ) set ) ;
let Y be ( ( ) ( ext-real-membered ) Subset of ( ( ) ( ) set ) ) ;
cluster Function-like V18(X : ( ( complex-membered ) ( complex-membered ) set ) ,Y : ( ( V35() ) ( V35() ) set ) ) bounded -> Function-like V18(X : ( ( complex-membered ) ( complex-membered ) set ) ,Y : ( ( V35() ) ( V35() ) set ) ) bounded_above bounded_below for ( ( ) ( ) Element of K19(K20(X : ( ( complex-membered ) ( complex-membered ) set ) ,Y : ( ( V35() ) ( V35() ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;
cluster Function-like V18(X : ( ( complex-membered ) ( complex-membered ) set ) ,Y : ( ( V35() ) ( V35() ) set ) ) bounded_above bounded_below -> Function-like V18(X : ( ( complex-membered ) ( complex-membered ) set ) ,Y : ( ( V35() ) ( V35() ) set ) ) bounded for ( ( ) ( ) Element of K19(K20(X : ( ( complex-membered ) ( complex-membered ) set ) ,Y : ( ( V35() ) ( V35() ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;
end;

theorem :: SUPINF_2:21
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) holds
( F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded iff ( sup F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) < +infty : ( ( ) ( V36() ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) & -infty : ( ( ) ( V36() ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) < inf F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ) ;

theorem :: SUPINF_2:22
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) holds
( F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded_above iff - F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded_below ) ;

theorem :: SUPINF_2:23
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) holds
( F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded_below iff - F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded_above ) ;

theorem :: SUPINF_2:24
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) holds
( F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded iff - F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) , - b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded ) ;

theorem :: SUPINF_2:25
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) )
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) holds
( -infty : ( ( ) ( V36() ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) <= F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) R_eal) & F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) R_eal) <= +infty : ( ( ) ( V36() ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ;

theorem :: SUPINF_2:26
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) )
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) st Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) c= REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) holds
( -infty : ( ( ) ( V36() ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) < F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) R_eal) & F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) R_eal) < +infty : ( ( ) ( V36() ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ;

theorem :: SUPINF_2:27
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) )
for x being ( ( ) ( ) Element of X : ( ( non empty ) ( non empty ) set ) ) holds
( inf F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) <= F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) R_eal) & F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ext-real ) R_eal) <= sup F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ;

theorem :: SUPINF_2:28
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) st Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) c= REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) holds
( F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded_above iff sup F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) in REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) ;

theorem :: SUPINF_2:29
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) st Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) c= REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) holds
( F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded_below iff inf F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) in REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) ;

theorem :: SUPINF_2:30
for X being ( ( non empty ) ( non empty ) set )
for Y being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) st Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) c= REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) holds
( F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded iff ( inf F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) in REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) & sup F : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) in REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) ) ;

theorem :: SUPINF_2:31
for X being ( ( non empty ) ( non empty ) set )
for Y, Z being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F1 being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) )
for F2 being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Z : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) st F1 : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded_above & F2 : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded_above holds
F1 : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) + F2 : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded_above ;

theorem :: SUPINF_2:32
for X being ( ( non empty ) ( non empty ) set )
for Y, Z being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F1 being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) )
for F2 being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Z : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) st F1 : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded_below & F2 : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded_below holds
F1 : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) + F2 : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded_below ;

theorem :: SUPINF_2:33
for X being ( ( non empty ) ( non empty ) set )
for Y, Z being ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) )
for F1 being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Y : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) )
for F2 being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of X : ( ( non empty ) ( non empty ) set ) ,Z : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) st F1 : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded & F2 : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded holds
F1 : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) + F2 : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) -valued Function-like non empty total V18(b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ext-real-valued ) Function of b1 : ( ( non empty ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) + b3 : ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) is bounded ;

theorem :: SUPINF_2:34
( incl NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) -valued NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -valued RAT : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered rational-membered V46() ) set ) -valued INT : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered rational-membered integer-membered V46() ) set ) -valued Function-like one-to-one non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) complex-valued ext-real-valued real-valued natural-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( Relation-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) & incl NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) -valued NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -valued RAT : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered rational-membered V46() ) set ) -valued INT : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered rational-membered integer-membered V46() ) set ) -valued Function-like one-to-one non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) complex-valued ext-real-valued real-valued natural-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( Relation-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is one-to-one & NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) = rng (incl NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) -valued NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -valued RAT : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered rational-membered V46() ) set ) -valued INT : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered rational-membered integer-membered V46() ) set ) -valued Function-like one-to-one non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) complex-valued ext-real-valued real-valued natural-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( Relation-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) & rng (incl NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) -valued NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -valued RAT : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered rational-membered V46() ) set ) -valued INT : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered rational-membered integer-membered V46() ) set ) -valued Function-like one-to-one non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) complex-valued ext-real-valued real-valued natural-valued ) Element of K19(K20(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( Relation-like complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) is ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) ) ;

definition
let D be ( ( non empty ) ( non empty ) set ) ;
let IT be ( ( ) ( ) Subset of ( ( ) ( ) set ) ) ;
:: original: countable
redefine attr IT is countable means :: SUPINF_2:def 8
( IT : ( ( ) ( ) set ) is empty or ex F being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ,D : ( ( ) ( ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined D : ( ( ) ( ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ,D : ( ( ) ( ) set ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ,D : ( ( ) ( ) set ) ) st IT : ( ( ) ( ) set ) = rng F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ,D : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined D : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ,D : ( ( non empty ) ( non empty ) set ) ) ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ,D : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of K19(D : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) );
end;

registration
cluster non empty ext-real-membered V78() for ( ( ) ( ) Element of K19(ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
mode Denum_Set_of_R_EAL is ( ( non empty V78() ) ( non empty ext-real-membered V78() ) Subset of ( ( ) ( ) set ) ) ;
end;

definition
let IT be ( ( ) ( ) set ) ;
attr IT is nonnegative means :: SUPINF_2:def 9
 for x being ( ( ) ( ext-real ) R_eal) st x : ( ( ) ( ext-real ) R_eal) in IT : ( ( ) ( ) set ) holds
0. : ( ( ) ( ext-real ) R_eal) <= x : ( ( ) ( ext-real ) R_eal) ;
end;

registration
cluster non empty ext-real-membered countable nonnegative for ( ( ) ( ) Element of K19(ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
mode Pos_Denum_Set_of_R_EAL is ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Denum_Set_of_R_EAL) ;
end;

definition
let D be ( ( non empty countable ) ( non empty ext-real-membered countable ) Denum_Set_of_R_EAL) ;
mode Num of D -> ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) means :: SUPINF_2:def 10
D : ( ( ) ( ) set ) = rng it : ( ( ) ( ) set ) : ( ( ) ( ext-real-membered ) Element of K19(ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
let D be ( ( non empty countable ) ( non empty ext-real-membered countable ) Denum_Set_of_R_EAL) ;
let N be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of D : ( ( non empty countable ) ( non empty ext-real-membered countable ) Denum_Set_of_R_EAL) ) ;
func Ser (D,N) -> ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) means :: SUPINF_2:def 11
( it : ( ( ) ( ) set ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) = N : ( ( ) ( ) set ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) )
for y being ( ( ) ( ext-real ) R_eal) st y : ( ( ) ( ext-real ) R_eal) = it : ( ( ) ( ) set ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) holds
it : ( ( ) ( ) set ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) = y : ( ( ) ( ext-real ) R_eal) + (N : ( ( ) ( ) set ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) ) );
end;

theorem :: SUPINF_2:35
for D being ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL)
for N being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of D : ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) )
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) holds 0. : ( ( ) ( ext-real ) R_eal) <= N : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of b1 : ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) ;

theorem :: SUPINF_2:36
for D being ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL)
for N being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of D : ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) )
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (Ser (D : ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) ,N : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of b1 : ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) ) )) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) <= (Ser (D : ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) ,N : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of b1 : ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) ) )) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) & 0. : ( ( ) ( ext-real ) R_eal) <= (Ser (D : ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) ,N : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of b1 : ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) ) )) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) ) ;

theorem :: SUPINF_2:37
for D being ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL)
for N being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of D : ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) )
for n, m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Ser (D : ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) ,N : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of b1 : ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) ) )) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) <= (Ser (D : ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) ,N : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of b1 : ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) ) )) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) + m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) ;

definition
let D be ( ( non empty countable ) ( non empty ext-real-membered countable ) Denum_Set_of_R_EAL) ;
mode Set_of_Series of D -> ( ( non empty ) ( non empty ext-real-membered ) Subset of ( ( ) ( ) set ) ) means :: SUPINF_2:def 12
 ex N being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of D : ( ( ) ( ) set ) ) st it : ( ( ) ( ) set ) = rng (Ser (D : ( ( ) ( ) set ) ,N : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of D : ( ( non empty countable ) ( non empty ext-real-membered countable ) Denum_Set_of_R_EAL) ) )) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( ) ( non empty ext-real-membered ) Element of K19(ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
let D be ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) ;
let N be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of D : ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) ) ;
func SUM (D,N) -> ( ( ) ( ext-real ) R_eal) equals :: SUPINF_2:def 13
 sup (rng (Ser (D : ( ( ) ( ) set ) ,N : ( ( ) ( ) set ) )) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) : ( ( ) ( non empty ext-real-membered ) Element of K19(ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) R_eal) ;
end;

definition
let D be ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) ;
let N be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of D : ( ( non empty countable nonnegative ) ( non empty ext-real-membered countable nonnegative ) Pos_Denum_Set_of_R_EAL) ) ;
pred D is_sumable N means :: SUPINF_2:def 14
 SUM (D : ( ( ) ( ) set ) ,N : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) R_eal) in REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ;
end;

theorem :: SUPINF_2:38
for F being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) holds rng F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( ) ( non empty ext-real-membered ) Element of K19(ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( ) ( ) set ) ) is ( ( non empty countable ) ( non empty ext-real-membered countable ) Denum_Set_of_R_EAL) ;

definition
let F be ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ;
:: original: rng
redefine func rng F -> ( ( non empty countable ) ( non empty ext-real-membered countable ) Denum_Set_of_R_EAL) ;
end;

definition
let F be ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ;
func Ser F -> ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) means :: SUPINF_2:def 15
 for N being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of rng F : ( ( ) ( ) set ) : ( ( non empty countable ) ( non empty ext-real-membered countable ) Denum_Set_of_R_EAL) ) st N : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of rng F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( non empty countable ) ( non empty ext-real-membered countable ) Denum_Set_of_R_EAL) ) = F : ( ( ) ( ) set ) holds
it : ( ( ) ( ) set ) = Ser ((rng F : ( ( ) ( ) set ) ) : ( ( non empty countable ) ( non empty ext-real-membered countable ) Denum_Set_of_R_EAL) ,N : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Num of rng F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( non empty countable ) ( non empty ext-real-membered countable ) Denum_Set_of_R_EAL) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ;
end;

definition
let R be ( ( Relation-like ) ( Relation-like ) Relation) ;
attr R is nonnegative means :: SUPINF_2:def 16
 rng R : ( ( ) ( ) set ) : ( ( ) ( ) set ) is nonnegative ;
end;

definition
let F be ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ;
func SUM F -> ( ( ) ( ext-real ) R_eal) equals :: SUPINF_2:def 17
 sup (rng (Ser F : ( ( ) ( ) set ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) : ( ( non empty countable ) ( non empty ext-real-membered countable ) Denum_Set_of_R_EAL) : ( ( ) ( ext-real ) R_eal) ;
end;

theorem :: SUPINF_2:39
for X being ( ( ) ( ) set )
for F being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) holds
( F : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is nonnegative iff for n being ( ( ) ( ) Element of X : ( ( ) ( ) set ) ) holds 0. : ( ( ) ( ext-real ) R_eal) <= F : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . n : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) : ( ( ) ( ext-real ) R_eal) ) ;

theorem :: SUPINF_2:40
for F being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) )
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) st F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) is nonnegative holds
( (Ser F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) <= (Ser F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) & 0. : ( ( ) ( ext-real ) R_eal) <= (Ser F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) ) ;

theorem :: SUPINF_2:41
for F being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) st F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) is nonnegative holds
for n, m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Ser F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) <= (Ser F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) + m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) ;

theorem :: SUPINF_2:42
for F1, F2 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) holds F1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) <= F2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) ) holds
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Ser F1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) <= (Ser F2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) ;

theorem :: SUPINF_2:43
for F1, F2 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) holds F1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) <= F2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) ) holds
SUM F1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( ) ( ext-real ) R_eal) <= SUM F2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( ) ( ext-real ) R_eal) ;

theorem :: SUPINF_2:44
for F being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) holds
( (Ser F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) = F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) )
for y being ( ( ) ( ext-real ) R_eal) st y : ( ( ) ( ext-real ) R_eal) = (Ser F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) holds
(Ser F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) = y : ( ( ) ( ext-real ) R_eal) + (F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . (n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ext-real ) R_eal) : ( ( ) ( ext-real ) R_eal) ) ) ;

theorem :: SUPINF_2:45
for F being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) st F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) is nonnegative & ex n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) st F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) = +infty : ( ( ) ( V36() ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) holds
SUM F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( ) ( ext-real ) R_eal) = +infty : ( ( ) ( V36() ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ;

definition
let F be ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ;
attr F is summable means :: SUPINF_2:def 18
 SUM F : ( ( ) ( ) set ) : ( ( ) ( ext-real ) R_eal) in REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ;
end;

theorem :: SUPINF_2:46
for F being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) st F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) is nonnegative & ex n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) st F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) = +infty : ( ( ) ( V36() ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) holds
not F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) is summable ;

theorem :: SUPINF_2:47
for F1, F2 being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) st F1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) is nonnegative & ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) holds F1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) <= F2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) ) & F2 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) is summable holds
F1 : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) is summable ;

theorem :: SUPINF_2:48
for F being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) st F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) is nonnegative holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real ) Nat) st ( for r being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real ) Nat) <= r : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) holds
F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . r : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) = 0. : ( ( ) ( ext-real ) R_eal) ) holds
SUM F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) : ( ( ) ( ext-real ) R_eal) = (Ser F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real ) Nat) : ( ( ) ( ext-real ) R_eal) ;

theorem :: SUPINF_2:49
for F being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) st ( for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) holds F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) in REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) holds
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Ser F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) in REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ;

theorem :: SUPINF_2:50
for F being ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) st F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) is nonnegative & ex n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) st
( ( for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) holds
F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) = 0. : ( ( ) ( ext-real ) R_eal) ) & ( for k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) st k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) holds
F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) . k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V35() V36() ext-real complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V50() V51() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ext-real ) R_eal) <> +infty : ( ( ) ( V36() ext-real ) Element of ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ) holds
F : ( ( Function-like V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like non empty total V18( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) ext-real-valued ) Function of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V46() ) Element of K19(REAL : ( ( ) ( non empty V12() V23() complex-membered ext-real-membered real-membered V46() ) set ) ) : ( ( ) ( ) set ) ) , ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) ) is summable ;

theorem :: SUPINF_2:51
for X being ( ( ) ( ) set )
for F being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) holds
( F : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is nonnegative iff for n being ( ( ) ( ) set ) holds 0. : ( ( ) ( ext-real ) R_eal) <= F : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . n : ( ( ) ( ) set ) : ( ( ) ( ext-real ) R_eal) ) ;

theorem :: SUPINF_2:52
for X being ( ( ) ( ) set )
for F being ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) st ( for n being ( ( ) ( ) set ) st n : ( ( ) ( ) set ) in dom F : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) : ( ( ) ( ) Element of K19(b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
0. : ( ( ) ( ext-real ) R_eal) <= F : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) . n : ( ( ) ( ) set ) : ( ( ) ( ext-real ) R_eal) ) holds
F : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined ExtREAL : ( ( ) ( non empty ext-real-membered ) set ) -valued Function-like ext-real-valued ) PartFunc of ,) is nonnegative ;