:: REAL_3 semantic presentation

begin

registration
let n be ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ;
cluster n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) div 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( integer ) ( complex real ext-real integer rational ) set ) -> zero integer ;
cluster n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) mod 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( integer ) ( complex real ext-real integer rational ) set ) -> zero integer ;
cluster 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) div n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) : ( ( integer ) ( complex real ext-real integer rational ) set ) -> zero integer ;
cluster 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) mod n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) : ( ( integer ) ( complex real ext-real integer rational ) set ) -> zero integer ;
end;

registration
let c be ( ( complex ) ( complex ) number ) ;
cluster c : ( ( complex ) ( complex ) set ) - c : ( ( complex ) ( complex ) set ) : ( ( ) ( ) set ) -> zero ;
cluster c : ( ( complex ) ( complex ) set ) / 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) -> zero ;
end;

registration
cluster [\0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) /] : ( ( integer ) ( complex real ext-real integer rational ) set ) -> zero integer ;
end;

theorem :: REAL_3:1
for r being ( ( real ) ( complex real ext-real ) number ) st 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( real ) ( complex real ext-real ) number ) & r : ( ( real ) ( complex real ext-real ) number ) < 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) < 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / r : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:2
for i being ( ( integer ) ( complex real ext-real integer rational ) Integer)
for r being ( ( real ) ( complex real ext-real ) number ) st i : ( ( integer ) ( complex real ext-real integer rational ) Integer) <= r : ( ( real ) ( complex real ext-real ) number ) & r : ( ( real ) ( complex real ext-real ) number ) < i : ( ( integer ) ( complex real ext-real integer rational ) Integer) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) holds
[\r : ( ( real ) ( complex real ext-real ) number ) /] : ( ( integer ) ( complex real ext-real integer rational ) set ) = i : ( ( integer ) ( complex real ext-real integer rational ) Integer) ;

theorem :: REAL_3:3
for m, n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds [\(m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) / n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real non negative rational ) Element of COMPLEX : ( ( ) ( non zero V45() V70() V76() ) set ) ) /] : ( ( integer ) ( complex real ext-real integer rational ) set ) = m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) div n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:4
for m, n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) mod n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) / n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real non negative rational ) Element of COMPLEX : ( ( ) ( non zero V45() V70() V76() ) set ) ) = m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) div n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:5
for m, n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) / n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real non negative rational ) Element of COMPLEX : ( ( ) ( non zero V45() V70() V76() ) set ) ) = m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) div n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) mod n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:6
for m, n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds frac (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) / n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real non negative rational ) Element of COMPLEX : ( ( ) ( non zero V45() V70() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) mod n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real non negative rational ) Element of COMPLEX : ( ( ) ( non zero V45() V70() V76() ) set ) ) ;

theorem :: REAL_3:7
for p being ( ( rational ) ( complex real ext-real rational ) number ) st p : ( ( rational ) ( complex real ext-real rational ) number ) >= 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex m, n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st
( n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( rational ) ( complex real ext-real rational ) number ) = m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) / n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real non negative rational ) Element of COMPLEX : ( ( ) ( non zero V45() V70() V76() ) set ) ) ) ;

registration
cluster non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() for ( ( ) ( ) Element of K6(K7(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
mode Integer_Sequence is ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) ;
end;

theorem :: REAL_3:8
for f being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) holds
( f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) is ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) iff ( dom f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) = NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) is integer ) ) ) ;

theorem :: REAL_3:9
for f being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) holds
( f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) is ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) ) V35() V36() V37() ) Function of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) ) iff f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) is ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) ) ;

theorem :: REAL_3:10
for f being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) holds
( f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) is ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) iff ( dom f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) = NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) . x : ( ( ) ( ) set ) : ( ( ) ( ) set ) is natural ) ) ) ;

theorem :: REAL_3:11
for f being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) holds
( f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) is ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) Function of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) iff f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) is ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

begin

definition
let m, n be ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ;
func modSeq (m,n) -> ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) means :: REAL_3:def 1
( it : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = m : ( ( ) ( ) set ) mod n : ( ( ) ( ) set ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) & it : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( ) ( ) set ) mod (m : ( ( ) ( ) set ) mod n : ( ( ) ( ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds it : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) . (k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = (it : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) mod (it : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) . (k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) );
end;

definition
let m, n be ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ;
func divSeq (m,n) -> ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) means :: REAL_3:def 2
( it : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = m : ( ( ) ( ) set ) div n : ( ( ) ( ) set ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) & it : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( ) ( ) set ) div (m : ( ( ) ( ) set ) mod n : ( ( ) ( ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds it : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) set ) . (k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = ((modSeq (m : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) div ((modSeq (m : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . (k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) );
end;

theorem :: REAL_3:12
for m, n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (divSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) div ((modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:13
for m, n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) mod ((modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:14
for a, b, m, n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) & (modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:15
for a, b, m, n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
( not a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) < b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) or (modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > (modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) or (modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: REAL_3:16
for m, n, a being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st (divSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . (a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:17
for a, b, m, n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) & a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) & (divSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(divSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:18
for a, b, m, n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) < b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) & (modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(divSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:19
for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) = (((divSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) + ((modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:20
for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) / n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real non negative rational ) Element of COMPLEX : ( ( ) ( non zero V45() V70() V76() ) set ) ) = ((divSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) + (1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) / ((modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real non negative rational ) Element of COMPLEX : ( ( ) ( non zero V45() V70() V76() ) set ) ) ) : ( ( ) ( complex real ext-real non negative rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real non negative rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:21
for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds divSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) --> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero T-Sequence-like Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined RAT : ( ( ) ( non zero V45() V70() V71() V72() V73() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) Element of K6(K7(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ,NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( RAT : ( ( ) ( non zero V45() V70() V71() V72() V73() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued V35() V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: REAL_3:22
for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) --> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero T-Sequence-like Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined RAT : ( ( ) ( non zero V45() V70() V71() V72() V73() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) Element of K6(K7(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ,NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( RAT : ( ( ) ( non zero V45() V70() V71() V72() V73() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued V35() V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: REAL_3:23
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds divSeq (0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) --> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero T-Sequence-like Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined RAT : ( ( ) ( non zero V45() V70() V71() V72() V73() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) Element of K6(K7(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ,NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( RAT : ( ( ) ( non zero V45() V70() V71() V72() V73() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued V35() V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: REAL_3:24
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds modSeq (0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) --> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero T-Sequence-like Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined RAT : ( ( ) ( non zero V45() V70() V71() V72() V73() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) Element of K6(K7(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ,NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( RAT : ( ( ) ( non zero V45() V70() V71() V72() V73() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued V35() V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: REAL_3:25
for m, n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ex k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st
( (divSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) & (modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

begin

definition
let r be ( ( real ) ( complex real ext-real ) number ) ;
func remainders_for_scf r -> ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) means :: REAL_3:def 3
( it : ( ( ) ( ) set ) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = r : ( ( ) ( ) set ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds it : ( ( ) ( ) set ) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / (frac (it : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) );
end;

notation
let r be ( ( real ) ( complex real ext-real ) number ) ;
synonym rfs r for remainders_for_scf r;
end;

definition
let r be ( ( real ) ( complex real ext-real ) number ) ;
func SimpleContinuedFraction r -> ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) means :: REAL_3:def 4
 for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds it : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = [\((rfs r : ( ( ) ( ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) /] : ( ( integer ) ( complex real ext-real integer rational ) set ) ;
end;

notation
let r be ( ( real ) ( complex real ext-real ) number ) ;
synonym scf r for SimpleContinuedFraction r;
end;

theorem :: REAL_3:26
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) holds (rfs r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / (((rfs r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) - ((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:27
for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) st (rfs r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) & n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
(rfs r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:28
for n, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) st (rfs r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) & n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) <= m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
(scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:29
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for i being ( ( integer ) ( complex real ext-real integer rational ) Integer) holds (rfs i : ( ( integer ) ( complex real ext-real integer rational ) Integer) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:30
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for i being ( ( integer ) ( complex real ext-real integer rational ) Integer) holds
( (scf i : ( ( integer ) ( complex real ext-real integer rational ) Integer) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = i : ( ( integer ) ( complex real ext-real integer rational ) Integer) & (scf i : ( ( integer ) ( complex real ext-real integer rational ) Integer) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: REAL_3:31
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for i being ( ( integer ) ( complex real ext-real integer rational ) Integer) st i : ( ( integer ) ( complex real ext-real integer rational ) Integer) > 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (rfs (1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / i : ( ( integer ) ( complex real ext-real integer rational ) Integer) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = i : ( ( integer ) ( complex real ext-real integer rational ) Integer) & (rfs (1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / i : ( ( integer ) ( complex real ext-real integer rational ) Integer) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: REAL_3:32
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for i being ( ( integer ) ( complex real ext-real integer rational ) Integer) st i : ( ( integer ) ( complex real ext-real integer rational ) Integer) > 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (scf (1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / i : ( ( integer ) ( complex real ext-real integer rational ) Integer) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) & (scf (1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / i : ( ( integer ) ( complex real ext-real integer rational ) Integer) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = i : ( ( integer ) ( complex real ext-real integer rational ) Integer) & (scf (1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / i : ( ( integer ) ( complex real ext-real integer rational ) Integer) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: REAL_3:33
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
(rfs r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:34
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
r : ( ( real ) ( complex real ext-real ) number ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:35
for r being ( ( real ) ( complex real ext-real ) number ) holds frac r : ( ( real ) ( complex real ext-real ) number ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = r : ( ( real ) ( complex real ext-real ) number ) - ((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:36
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) holds (rfs r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = (rfs (1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / (frac r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:37
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = (scf (1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / (frac r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:38
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) >= 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) >= 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:39
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) >= 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) in NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: REAL_3:40
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) >= 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) & (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) <> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) >= 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:41
for m, n, k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
( (scf (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) / n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real non negative rational ) Element of COMPLEX : ( ( ) ( non zero V45() V70() V76() ) set ) ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = (divSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) & (rfs (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) / n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real non negative rational ) Element of COMPLEX : ( ( ) ( non zero V45() V70() V76() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) / ((modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real non negative rational ) Element of COMPLEX : ( ( ) ( non zero V45() V70() V76() ) set ) ) & (rfs (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) / n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real non negative rational ) Element of COMPLEX : ( ( ) ( non zero V45() V70() V76() ) set ) ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = ((modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((modSeq (m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ,n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) )) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V35() V36() V37() V38() ) sequence of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) . (k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real non negative rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ;

theorem :: REAL_3:42
for r being ( ( real ) ( complex real ext-real ) number ) holds
( r : ( ( real ) ( complex real ext-real ) number ) is rational iff ex n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st
for m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) >= n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds
(scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: REAL_3:43
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) <> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
r : ( ( real ) ( complex real ext-real ) number ) is irrational ;

begin

definition
let r be ( ( real ) ( complex real ext-real ) number ) ;
func convergent_numerators r -> ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) means :: REAL_3:def 5
( it : ( ( ) ( ) set ) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = (scf r : ( ( ) ( ) set ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) & it : ( ( ) ( ) set ) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = (((scf r : ( ( ) ( ) set ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((scf r : ( ( ) ( ) set ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds it : ( ( ) ( ) set ) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = (((scf r : ( ( ) ( ) set ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * (it : ( ( ) ( ) set ) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) + (it : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) );
end;

definition
let r be ( ( real ) ( complex real ext-real ) number ) ;
func convergent_denominators r -> ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) means :: REAL_3:def 6
( it : ( ( ) ( ) set ) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) & it : ( ( ) ( ) set ) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = (scf r : ( ( ) ( ) set ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds it : ( ( ) ( ) set ) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = (((scf r : ( ( ) ( ) set ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * (it : ( ( ) ( ) set ) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) + (it : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) );
end;

notation
let r be ( ( real ) ( complex real ext-real ) number ) ;
synonym c_n r for convergent_numerators r;
synonym c_d r for convergent_denominators r;
end;

theorem :: REAL_3:44
for r being ( ( real ) ( complex real ext-real ) number ) st (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) in NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: REAL_3:45
for r being ( ( real ) ( complex real ext-real ) number ) st (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:46
for r being ( ( real ) ( complex real ext-real ) number ) st (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > ((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:47
for r being ( ( real ) ( complex real ext-real ) number )
for n1, n2 being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) = (c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) & n2 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) = (c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) holds
n1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) gcd n2 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:48
for r being ( ( real ) ( complex real ext-real ) number ) st (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) <> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) >= tau : ( ( real ) ( complex real ext-real ) set ) |^ n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) set ) ;

theorem :: REAL_3:49
for b being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) st (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) <= b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) <= ((b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + (sqrt ((b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ^2) : ( ( ) ( complex real ext-real ) set ) + 4 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) |^ (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:50
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) holds (c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) in NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: REAL_3:51
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) holds (c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) >= 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:52
for r being ( ( real ) ( complex real ext-real ) number ) st (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:53
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) holds (c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) >= ((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:54
for r being ( ( real ) ( complex real ext-real ) number ) st (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > ((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:55
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) >= 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / (((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) < 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / (((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * (((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:56
for b being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) <= b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) <= ((b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + (sqrt ((b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ^2) : ( ( ) ( complex real ext-real ) set ) + 4 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) |^ (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:57
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number )
for n1, n2 being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) = (c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) & n2 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) = (c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) holds
n1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) gcd n2 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) = 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: REAL_3:58
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) >= 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / ((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:59
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) <= (2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * ((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:60
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) <> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / (((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * (((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) <= 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / (((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ^2) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:61
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) <> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) >= tau : ( ( real ) ( complex real ext-real ) set ) |^ n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) set ) ;

theorem :: REAL_3:62
for a being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) st a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) >= a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) >= ((a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + (sqrt ((a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ^2) : ( ( ) ( complex real ext-real ) set ) + 4 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) |^ n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:63
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) holds ((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = ((((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) + ((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) + ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:64
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) holds (((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) - (((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = (- 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real non positive integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) |^ n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:65
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) <> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
(((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) - (((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = ((- 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real non positive integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) |^ n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / (((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:66
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) holds (((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) - (((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = ((- 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real non positive integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) |^ n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:67
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat)
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) <> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
(((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) - (((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = (((- 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real non positive integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) |^ n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / (((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:68
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) <> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) >= 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = (((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) - ((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) - 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / (((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) - ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) - 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:69
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) <> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds abs ((((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) - (((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / (abs (((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) * ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:70
for r being ( ( real ) ( complex real ext-real ) number ) st (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds ((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . ((2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . ((2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > ((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

definition
let r be ( ( real ) ( complex real ext-real ) number ) ;
func convergents_of_continued_fractions r -> ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) equals :: REAL_3:def 7
(c_n r : ( ( ) ( ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) /" (c_d r : ( ( ) ( ) set ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) : ( ( Function-like ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Element of K6(K7(NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( V35() V36() V37() ) set ) ) : ( ( ) ( ) set ) ) ;
end;

notation
let r be ( ( real ) ( complex real ext-real ) number ) ;
synonym cocf r for convergents_of_continued_fractions r;
end;

theorem :: REAL_3:71
for r being ( ( real ) ( complex real ext-real ) number ) holds (cocf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:72
for r being ( ( real ) ( complex real ext-real ) number ) st (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) <> 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(cocf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = ((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) + (1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / ((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:73
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
(cocf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = ((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) + (1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / (((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) + (1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / ((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:74
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
(cocf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . 3 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = ((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) + (1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / (((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) + (1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / (((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) + (1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / ((scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 3 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:75
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) >= 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . ((2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . ((2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) < ((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . ((2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . ((2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:76
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) >= 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > ((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . ((2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . ((2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

theorem :: REAL_3:77
for r being ( ( real ) ( complex real ext-real ) number ) st ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) >= 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) < ((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . ((2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_d r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . ((2 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;

definition
let r be ( ( real ) ( complex real ext-real ) number ) ;
func backContinued_fraction r -> ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) means :: REAL_3:def 8
( it : ( ( ) ( ) set ) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = (scf r : ( ( ) ( ) set ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) & ( for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds it : ( ( ) ( ) set ) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = (1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) / (it : ( ( ) ( ) set ) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) + ((scf r : ( ( ) ( ) set ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) );
end;

notation
let r be ( ( real ) ( complex real ext-real ) number ) ;
synonym bcf r for backContinued_fraction r;
end;

theorem :: REAL_3:78
for r being ( ( real ) ( complex real ext-real ) number ) st (scf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued INT : ( ( ) ( non zero V45() V70() V71() V72() V73() V74() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Integer_Sequence) . 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real integer rational ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) > 0 : ( ( ) ( zero epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real ext-real non positive non negative integer rational V70() V71() V72() V73() V74() V75() V76() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) holds
for n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) holds (bcf r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) = ((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) + 1 : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real ext-real positive non negative integer rational V70() V71() V72() V73() V74() V75() ) Element of NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) / ((c_n r : ( ( real ) ( complex real ext-real ) number ) ) : ( ( Function-like V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ) ( non zero Relation-like NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) -valued Function-like V26( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) ) V30( NAT : ( ( ) ( non zero epsilon-transitive epsilon-connected ordinal V70() V71() V72() V73() V74() V75() V76() ) Element of K6(REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) V35() V36() V37() ) Real_Sequence) . n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real ext-real non negative integer rational ) Nat) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) : ( ( ) ( complex real ext-real ) Element of REAL : ( ( ) ( non zero V45() V70() V71() V72() V76() ) set ) ) ;