:: INT_4 semantic presentation

begin

theorem :: INT_4:1
for X being ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set )
for a being ( ( real ) ( complex real ext-real ) number ) holds X : ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set ) ,a : ( ( real ) ( complex real ext-real ) number ) ++ X : ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set ) : ( ( ) ( complex-membered ext-real-membered real-membered ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) are_equipotent ;

registration
let X be ( ( finite real-membered ) ( finite complex-membered ext-real-membered real-membered ) set ) ;
let a be ( ( real ) ( complex real ext-real ) number ) ;
cluster K687(X : ( ( finite real-membered ) ( finite complex-membered ext-real-membered real-membered ) set ) ,a : ( ( real ) ( complex real ext-real ) set ) ) : ( ( ) ( ) set ) -> finite ;
end;

theorem :: INT_4:2
for X being ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set )
for a being ( ( real ) ( complex real ext-real ) number ) holds card X : ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set ) : ( ( cardinal ) ( epsilon-transitive epsilon-connected ordinal cardinal ) set ) = card (a : ( ( real ) ( complex real ext-real ) number ) ++ X : ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set ) ) : ( ( ) ( complex-membered ext-real-membered real-membered ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) : ( ( cardinal ) ( epsilon-transitive epsilon-connected ordinal cardinal ) set ) ;

theorem :: INT_4:3
for X being ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set )
for a being ( ( real ) ( complex real ext-real ) number ) st a : ( ( real ) ( complex real ext-real ) number ) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
X : ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set ) ,a : ( ( real ) ( complex real ext-real ) number ) ** X : ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set ) : ( ( ) ( complex-membered ext-real-membered real-membered ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) are_equipotent ;

theorem :: INT_4:4
for X being ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set )
for a being ( ( real ) ( complex real ext-real ) number ) holds
( ( a : ( ( real ) ( complex real ext-real ) number ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & not X : ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set ) is empty implies a : ( ( real ) ( complex real ext-real ) number ) ** X : ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set ) : ( ( ) ( complex-membered ext-real-membered real-membered ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) = {0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) } : ( ( ) ( non empty V2() finite V38() 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) set ) ) & ( not a : ( ( real ) ( complex real ext-real ) number ) ** X : ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set ) : ( ( ) ( complex-membered ext-real-membered real-membered ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) = {0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) } : ( ( ) ( non empty V2() finite V38() 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) set ) or a : ( ( real ) ( complex real ext-real ) number ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) or X : ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set ) = {0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) } : ( ( ) ( non empty V2() finite V38() 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) set ) ) ) ;

registration
let X be ( ( finite real-membered ) ( finite complex-membered ext-real-membered real-membered ) set ) ;
let a be ( ( real ) ( complex real ext-real ) number ) ;
cluster K693(X : ( ( finite real-membered ) ( finite complex-membered ext-real-membered real-membered ) set ) ,a : ( ( real ) ( complex real ext-real ) set ) ) : ( ( ) ( ) set ) -> finite ;
end;

theorem :: INT_4:5
for X being ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set )
for a being ( ( real ) ( complex real ext-real ) number ) st a : ( ( real ) ( complex real ext-real ) number ) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
card X : ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set ) : ( ( cardinal ) ( epsilon-transitive epsilon-connected ordinal cardinal ) set ) = card (a : ( ( real ) ( complex real ext-real ) number ) ** X : ( ( real-membered ) ( complex-membered ext-real-membered real-membered ) set ) ) : ( ( ) ( complex-membered ext-real-membered real-membered ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) : ( ( cardinal ) ( epsilon-transitive epsilon-connected ordinal cardinal ) set ) ;

begin

theorem :: INT_4:6
for i, i1 being ( ( integer ) ( complex real integer ext-real ) Integer) st i : ( ( integer ) ( complex real integer ext-real ) Integer) divides i1 : ( ( integer ) ( complex real integer ext-real ) Integer) & i1 : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
abs i : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) <= abs i1 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:7
for i1, i2, i3 being ( ( integer ) ( complex real integer ext-real ) Integer) st i3 : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
( i1 : ( ( integer ) ( complex real integer ext-real ) Integer) divides i2 : ( ( integer ) ( complex real integer ext-real ) Integer) iff i1 : ( ( integer ) ( complex real integer ext-real ) Integer) * i3 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( complex real integer ext-real ) set ) divides i2 : ( ( integer ) ( complex real integer ext-real ) Integer) * i3 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( complex real integer ext-real ) set ) ) ;

theorem :: INT_4:8
for a, b, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat)
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) mod m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) mod m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
(a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) |^ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) mod m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = (b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) |^ n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) mod m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:9
for i1, i, i2, i3 being ( ( integer ) ( complex real integer ext-real ) Integer) st i1 : ( ( integer ) ( complex real integer ext-real ) Integer) * i : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( complex real integer ext-real ) set ) ,i2 : ( ( integer ) ( complex real integer ext-real ) Integer) * i : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( complex real integer ext-real ) set ) are_congruent_mod i3 : ( ( integer ) ( complex real integer ext-real ) Integer) & i : ( ( integer ) ( complex real integer ext-real ) Integer) ,i3 : ( ( integer ) ( complex real integer ext-real ) Integer) are_relative_prime holds
i1 : ( ( integer ) ( complex real integer ext-real ) Integer) ,i2 : ( ( integer ) ( complex real integer ext-real ) Integer) are_congruent_mod i3 : ( ( integer ) ( complex real integer ext-real ) Integer) ;

theorem :: INT_4:10
for i1, i2, i3 being ( ( integer ) ( complex real integer ext-real ) Integer)
for k being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) st i1 : ( ( integer ) ( complex real integer ext-real ) Integer) ,i2 : ( ( integer ) ( complex real integer ext-real ) Integer) are_congruent_mod i3 : ( ( integer ) ( complex real integer ext-real ) Integer) holds
i1 : ( ( integer ) ( complex real integer ext-real ) Integer) * k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( ) ( complex real integer ext-real ) set ) ,i2 : ( ( integer ) ( complex real integer ext-real ) Integer) * k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( ) ( complex real integer ext-real ) set ) are_congruent_mod i3 : ( ( integer ) ( complex real integer ext-real ) Integer) * k : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( ) ( complex real integer ext-real ) set ) ;

theorem :: INT_4:11
for i1, i2, i, i3 being ( ( integer ) ( complex real integer ext-real ) Integer) st i1 : ( ( integer ) ( complex real integer ext-real ) Integer) ,i2 : ( ( integer ) ( complex real integer ext-real ) Integer) are_congruent_mod i : ( ( integer ) ( complex real integer ext-real ) Integer) holds
i1 : ( ( integer ) ( complex real integer ext-real ) Integer) * i3 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( complex real integer ext-real ) set ) ,i2 : ( ( integer ) ( complex real integer ext-real ) Integer) * i3 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( complex real integer ext-real ) set ) are_congruent_mod i : ( ( integer ) ( complex real integer ext-real ) Integer) ;

theorem :: INT_4:12
for i being ( ( integer ) ( complex real integer ext-real ) Integer) holds 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) mod i : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) ;

theorem :: INT_4:13
for b being ( ( integer ) ( complex real integer ext-real ) Integer) st b : ( ( integer ) ( complex real integer ext-real ) Integer) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
for a being ( ( integer ) ( complex real integer ext-real ) Integer) ex q, r being ( ( integer ) ( complex real integer ext-real ) Integer) st
( a : ( ( integer ) ( complex real integer ext-real ) Integer) = (b : ( ( integer ) ( complex real integer ext-real ) Integer) * q : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) + r : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( complex real integer ext-real ) set ) & r : ( ( integer ) ( complex real integer ext-real ) Integer) >= 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & r : ( ( integer ) ( complex real integer ext-real ) Integer) < b : ( ( integer ) ( complex real integer ext-real ) Integer) ) ;

theorem :: INT_4:14
for i1, i2, i3 being ( ( integer ) ( complex real integer ext-real ) Integer) st i1 : ( ( integer ) ( complex real integer ext-real ) Integer) ,i2 : ( ( integer ) ( complex real integer ext-real ) Integer) are_congruent_mod i3 : ( ( integer ) ( complex real integer ext-real ) Integer) holds
i1 : ( ( integer ) ( complex real integer ext-real ) Integer) gcd i3 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) = i2 : ( ( integer ) ( complex real integer ext-real ) Integer) gcd i3 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ;

theorem :: INT_4:15
for a, m, b being ( ( integer ) ( complex real integer ext-real ) Integer) st a : ( ( integer ) ( complex real integer ext-real ) Integer) ,m : ( ( integer ) ( complex real integer ext-real ) Integer) are_relative_prime holds
ex x being ( ( integer ) ( complex real integer ext-real ) Integer) st ((a : ( ( integer ) ( complex real integer ext-real ) Integer) * x : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - b : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:16
for m, a, b being ( ( integer ) ( complex real integer ext-real ) Integer) st m : ( ( integer ) ( complex real integer ext-real ) Integer) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & a : ( ( integer ) ( complex real integer ext-real ) Integer) ,m : ( ( integer ) ( complex real integer ext-real ) Integer) are_relative_prime holds
ex n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) st ((a : ( ( integer ) ( complex real integer ext-real ) Integer) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ) : ( ( ) ( complex real integer ext-real ) set ) - b : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:17
for m, a, b being ( ( integer ) ( complex real integer ext-real ) Integer) st m : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & not a : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) divides b : ( ( integer ) ( complex real integer ext-real ) Integer) holds
for x being ( ( integer ) ( complex real integer ext-real ) Integer) holds not ((a : ( ( integer ) ( complex real integer ext-real ) Integer) * x : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - b : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:18
for m, a, b being ( ( integer ) ( complex real integer ext-real ) Integer) st m : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & a : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) divides b : ( ( integer ) ( complex real integer ext-real ) Integer) holds
ex x being ( ( integer ) ( complex real integer ext-real ) Integer) st ((a : ( ( integer ) ( complex real integer ext-real ) Integer) * x : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - b : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

begin

theorem :: INT_4:19
for n, p, q being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
(p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) * q : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) mod (p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) |^ n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) : ( ( integer ) ( complex real integer ext-real ) set ) = p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) * (q : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) mod (p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) |^ (n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) -' 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ;

theorem :: INT_4:20
for p, q, n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) holds (p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) * q : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) mod (p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) * n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) : ( ( integer ) ( complex real integer ext-real ) set ) = p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) * (q : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) mod n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ;

theorem :: INT_4:21
for n, p, q being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) is prime & p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ,q : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) are_relative_prime holds
not p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) divides q : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) mod (p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) |^ n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:22
for p, q, n being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
( (p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) - q : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ) : ( ( ) ( complex real integer ext-real ) set ) mod n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) iff p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) mod n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = q : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) mod n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) ;

theorem :: INT_4:23
for a, b, m being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) st m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
( a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) mod m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) mod m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) iff m : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) divides a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) - b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( ) ( complex real integer ext-real ) set ) ) ;

theorem :: INT_4:24
for n, p, q being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) is prime & p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ,q : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) are_relative_prime holds
for x being ( ( integer ) ( complex real integer ext-real ) Integer) holds not ((p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) * (x : ( ( integer ) ( complex real integer ext-real ) Integer) ^2) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) - q : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ) : ( ( ) ( complex real integer ext-real ) set ) mod (p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) |^ n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:25
for n, p, q being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) st n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) is prime & p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ,q : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) are_relative_prime holds
for x being ( ( integer ) ( complex real integer ext-real ) Integer) holds not ((p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) * x : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - q : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ) : ( ( ) ( complex real integer ext-real ) set ) mod (p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) |^ n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

begin

definition
let m be ( ( integer ) ( complex real integer ext-real ) Integer) ;
func Cong m -> ( ( ) ( Relation-like INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued V71() V72() V73() ) Relation of ) means :: INT_4:def 1
 for x, y being ( ( integer ) ( complex real integer ext-real ) Integer) holds
( [x : ( ( integer ) ( complex real integer ext-real ) Integer) ,y : ( ( integer ) ( complex real integer ext-real ) Integer) ] : ( ( ) ( ) set ) in it : ( ( ) ( ) set ) iff x : ( ( integer ) ( complex real integer ext-real ) Integer) ,y : ( ( integer ) ( complex real integer ext-real ) Integer) are_congruent_mod m : ( ( ) ( ) set ) );
end;

registration
let m be ( ( integer ) ( complex real integer ext-real ) Integer) ;
cluster Cong m : ( ( integer ) ( complex real integer ext-real ) set ) : ( ( ) ( Relation-like INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued V71() V72() V73() ) Relation of ) -> total ;
end;

registration
let m be ( ( integer ) ( complex real integer ext-real ) Integer) ;
cluster Cong m : ( ( integer ) ( complex real integer ext-real ) set ) : ( ( ) ( Relation-like INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued total V71() V72() V73() ) Relation of ) -> reflexive symmetric transitive ;
end;

theorem :: INT_4:26
for m, a, x, b being ( ( integer ) ( complex real integer ext-real ) Integer) st m : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & ((a : ( ( integer ) ( complex real integer ext-real ) Integer) * x : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - b : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
for y being ( ( integer ) ( complex real integer ext-real ) Integer) holds
( ( a : ( ( integer ) ( complex real integer ext-real ) Integer) ,m : ( ( integer ) ( complex real integer ext-real ) Integer) are_relative_prime & ((a : ( ( integer ) ( complex real integer ext-real ) Integer) * y : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - b : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) implies y : ( ( integer ) ( complex real integer ext-real ) Integer) in Class ((Cong m : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( Relation-like INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued total reflexive symmetric transitive V71() V72() V73() ) Relation of ) ,x : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered ) Element of K6(INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & ( y : ( ( integer ) ( complex real integer ext-real ) Integer) in Class ((Cong m : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( Relation-like INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued total reflexive symmetric transitive V71() V72() V73() ) Relation of ) ,x : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered ) Element of K6(INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) implies ((a : ( ( integer ) ( complex real integer ext-real ) Integer) * y : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - b : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) ) ;

theorem :: INT_4:27
for a, b, m, x being ( ( integer ) ( complex real integer ext-real ) Integer) st m : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & a : ( ( integer ) ( complex real integer ext-real ) Integer) ,m : ( ( integer ) ( complex real integer ext-real ) Integer) are_relative_prime & ((a : ( ( integer ) ( complex real integer ext-real ) Integer) * x : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - b : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
ex s being ( ( integer ) ( complex real integer ext-real ) Integer) st [x : ( ( integer ) ( complex real integer ext-real ) Integer) ,(b : ( ( integer ) ( complex real integer ext-real ) Integer) * s : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) ] : ( ( ) ( ) set ) in Cong m : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( Relation-like INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued total reflexive symmetric transitive V71() V72() V73() ) Relation of ) ;

theorem :: INT_4:28
for a, m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st a : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) > 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & a : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) are_relative_prime holds
for b, x being ( ( integer ) ( complex real integer ext-real ) Integer) st ((a : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) * x : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) Element of REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) - b : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) Element of REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) mod m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
[x : ( ( integer ) ( complex real integer ext-real ) Integer) ,(b : ( ( integer ) ( complex real integer ext-real ) Integer) * (a : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) |^ ((Euler m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) -' 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( complex real integer ext-real ) Element of REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) ] : ( ( ) ( ) set ) in Cong m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( Relation-like INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued total reflexive symmetric transitive V71() V72() V73() ) Relation of ) ;

theorem :: INT_4:29
for m, a, b being ( ( integer ) ( complex real integer ext-real ) Integer) st m : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & a : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) divides b : ( ( integer ) ( complex real integer ext-real ) Integer) holds
ex fp being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() ) FinSequence of INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) ) st
( len fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() ) FinSequence of INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = a : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) & ( for c being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st c : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() ) FinSequence of INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) holds
((a : ( ( integer ) ( complex real integer ext-real ) Integer) * (fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() ) FinSequence of INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) ) . c : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( complex real integer ext-real ) set ) ) : ( ( ) ( complex real integer ext-real ) set ) - b : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) & ( for c1, c2 being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st c1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() ) FinSequence of INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) & c2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() ) FinSequence of INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) & c1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) <> c2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
not fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() ) FinSequence of INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) ) . c1 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( complex real integer ext-real ) set ) ,fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() ) FinSequence of INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) ) . c2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( complex real integer ext-real ) set ) are_congruent_mod m : ( ( integer ) ( complex real integer ext-real ) Integer) ) ) ;

begin

theorem :: INT_4:30
for fp being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) )
for d, b, n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) & len fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) holds
Del ((fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ^ <*d : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) *> : ( ( ) ( non empty V2() Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -valued Function-like one-to-one constant finite 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) -element FinSequence-like FinSubsequence-like V71() V72() V73() V74() V75() V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ,b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = (Del (fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ,b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ^ <*d : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) *> : ( ( ) ( non empty V2() Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -valued Function-like one-to-one constant finite 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) -element FinSequence-like FinSubsequence-like V71() V72() V73() V74() V75() V76() V77() V78() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( non empty Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:31
for fp being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st len fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) >= 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & ( for b, c being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) & c : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) & b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) <> c : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
(fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) gcd (fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . c : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) holds
for b being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) holds
(Product (Del (fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ,b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) gcd (fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:32
for fp being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) )
for a being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) st a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) holds
fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) divides Product fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:33
for p being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat)
for fp being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) )
for a being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) st a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) & p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) divides fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
p : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) divides Product fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:34
for fp being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) )
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) )
for a being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) st len fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) & a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) >= 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) <= n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
(Del (fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ,a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . (len fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:35
for fp being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) )
for a being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat)
for b being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) & b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) & a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) <> b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & len fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) >= 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) divides Product (Del (fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ,a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:36
for fp being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) )
for a being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) st ( for b being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) st b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) holds
a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) divides fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) holds
a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) divides Sum fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:37
for fp being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st len fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) >= 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & ( for b, c being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st b : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) & c : ( ( integer ) ( complex real integer ext-real ) Integer) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) & b : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) <> c : ( ( integer ) ( complex real integer ext-real ) Integer) holds
(fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . b : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) gcd (fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . c : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) & ( for b being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st b : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) holds
fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . b : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) holds
for fp1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ex x being ( ( integer ) ( complex real integer ext-real ) Integer) st
for b being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) holds
(x : ( ( integer ) ( complex real integer ext-real ) Integer) - (fp1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( complex real integer ext-real ) Element of REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) mod (fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:38
for fp being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) )
for a being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) st ( for b, c being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) & c : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) & b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) <> c : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
(fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) gcd (fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . c : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) & ( for b being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) holds
fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) divides a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ) holds
Product fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) divides a : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ;

theorem :: INT_4:39
for x, y being ( ( integer ) ( complex real integer ext-real ) Integer)
for fp being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st ( for b, c being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st b : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) & c : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) & b : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) <> c : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
(fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . b : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) gcd (fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . c : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) & ( for b being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st b : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) holds
fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . b : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) > 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) holds
for fp1 being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st ( for b being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) holds
( (x : ( ( integer ) ( complex real integer ext-real ) Integer) - (fp1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( complex real integer ext-real ) Element of REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) mod (fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & (y : ( ( integer ) ( complex real integer ext-real ) Integer) - (fp1 : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( complex real integer ext-real ) Element of REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) mod (fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) . b : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) ) holds
x : ( ( integer ) ( complex real integer ext-real ) Integer) ,y : ( ( integer ) ( complex real integer ext-real ) Integer) are_congruent_mod Product fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() V74() ) FinSequence of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:40
for m1, m2, c1, c2 being ( ( integer ) ( complex real integer ext-real ) Integer) st m1 : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & m2 : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & m1 : ( ( integer ) ( complex real integer ext-real ) Integer) ,m2 : ( ( integer ) ( complex real integer ext-real ) Integer) are_relative_prime holds
ex r being ( ( integer ) ( complex real integer ext-real ) Integer) st
( ( for x being ( ( integer ) ( complex real integer ext-real ) Integer) st (x : ( ( integer ) ( complex real integer ext-real ) Integer) - c1 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m1 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & (x : ( ( integer ) ( complex real integer ext-real ) Integer) - c2 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m2 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
x : ( ( integer ) ( complex real integer ext-real ) Integer) ,c1 : ( ( integer ) ( complex real integer ext-real ) Integer) + (m1 : ( ( integer ) ( complex real integer ext-real ) Integer) * r : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) : ( ( ) ( complex real integer ext-real ) set ) are_congruent_mod m1 : ( ( integer ) ( complex real integer ext-real ) Integer) * m2 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( complex real integer ext-real ) set ) ) & ((m1 : ( ( integer ) ( complex real integer ext-real ) Integer) * r : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - (c2 : ( ( integer ) ( complex real integer ext-real ) Integer) - c1 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) ) : ( ( ) ( complex real integer ext-real ) set ) mod m2 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) ;

theorem :: INT_4:41
for m1, m2, c1, c2 being ( ( integer ) ( complex real integer ext-real ) Integer) st m1 : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & m2 : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & not m1 : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m2 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) divides c1 : ( ( integer ) ( complex real integer ext-real ) Integer) - c2 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( complex real integer ext-real ) set ) holds
for x being ( ( integer ) ( complex real integer ext-real ) Integer) holds
( not (x : ( ( integer ) ( complex real integer ext-real ) Integer) - c1 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m1 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) or not (x : ( ( integer ) ( complex real integer ext-real ) Integer) - c2 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m2 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) ;

theorem :: INT_4:42
for m1, m2, c2, c1 being ( ( integer ) ( complex real integer ext-real ) Integer) st m1 : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & m2 : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & m1 : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m2 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) divides c2 : ( ( integer ) ( complex real integer ext-real ) Integer) - c1 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( complex real integer ext-real ) set ) holds
ex r being ( ( integer ) ( complex real integer ext-real ) Integer) st
( ( for x being ( ( integer ) ( complex real integer ext-real ) Integer) st (x : ( ( integer ) ( complex real integer ext-real ) Integer) - c1 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m1 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & (x : ( ( integer ) ( complex real integer ext-real ) Integer) - c2 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m2 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
x : ( ( integer ) ( complex real integer ext-real ) Integer) ,c1 : ( ( integer ) ( complex real integer ext-real ) Integer) + (m1 : ( ( integer ) ( complex real integer ext-real ) Integer) * r : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) : ( ( ) ( complex real integer ext-real ) set ) are_congruent_mod m1 : ( ( integer ) ( complex real integer ext-real ) Integer) lcm m2 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) & (((m1 : ( ( integer ) ( complex real integer ext-real ) Integer) div (m1 : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m2 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) : ( ( integer ) ( complex real integer ext-real ) set ) * r : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - ((c2 : ( ( integer ) ( complex real integer ext-real ) Integer) - c1 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) div (m1 : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m2 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) : ( ( integer ) ( complex real integer ext-real ) set ) ) : ( ( ) ( complex real integer ext-real ) set ) mod (m2 : ( ( integer ) ( complex real integer ext-real ) Integer) div (m1 : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m2 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) : ( ( integer ) ( complex real integer ext-real ) set ) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) ;

theorem :: INT_4:43
for m1, m2, a, c1, b, c2 being ( ( integer ) ( complex real integer ext-real ) Integer) st m1 : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & m2 : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & a : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m1 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) divides c1 : ( ( integer ) ( complex real integer ext-real ) Integer) & b : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m2 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) divides c2 : ( ( integer ) ( complex real integer ext-real ) Integer) & m1 : ( ( integer ) ( complex real integer ext-real ) Integer) ,m2 : ( ( integer ) ( complex real integer ext-real ) Integer) are_relative_prime holds
ex w, r, s being ( ( integer ) ( complex real integer ext-real ) Integer) st
( ( for x being ( ( integer ) ( complex real integer ext-real ) Integer) st ((a : ( ( integer ) ( complex real integer ext-real ) Integer) * x : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - c1 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m1 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & ((b : ( ( integer ) ( complex real integer ext-real ) Integer) * x : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - c2 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m2 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
x : ( ( integer ) ( complex real integer ext-real ) Integer) ,r : ( ( integer ) ( complex real integer ext-real ) Integer) + ((m1 : ( ( integer ) ( complex real integer ext-real ) Integer) div (a : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m1 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) : ( ( integer ) ( complex real integer ext-real ) set ) * w : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) : ( ( ) ( complex real integer ext-real ) set ) are_congruent_mod (m1 : ( ( integer ) ( complex real integer ext-real ) Integer) div (a : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m1 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) : ( ( integer ) ( complex real integer ext-real ) set ) * (m2 : ( ( integer ) ( complex real integer ext-real ) Integer) div (b : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m2 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) : ( ( integer ) ( complex real integer ext-real ) set ) : ( ( ) ( complex real integer ext-real ) set ) ) & (((a : ( ( integer ) ( complex real integer ext-real ) Integer) div (a : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m1 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) : ( ( integer ) ( complex real integer ext-real ) set ) * r : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - (c1 : ( ( integer ) ( complex real integer ext-real ) Integer) div (a : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m1 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) : ( ( integer ) ( complex real integer ext-real ) set ) ) : ( ( ) ( complex real integer ext-real ) set ) mod (m1 : ( ( integer ) ( complex real integer ext-real ) Integer) div (a : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m1 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) : ( ( integer ) ( complex real integer ext-real ) set ) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & (((b : ( ( integer ) ( complex real integer ext-real ) Integer) div (b : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m2 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) : ( ( integer ) ( complex real integer ext-real ) set ) * s : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - (c2 : ( ( integer ) ( complex real integer ext-real ) Integer) div (b : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m2 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) : ( ( integer ) ( complex real integer ext-real ) set ) ) : ( ( ) ( complex real integer ext-real ) set ) mod (m2 : ( ( integer ) ( complex real integer ext-real ) Integer) div (b : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m2 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) : ( ( integer ) ( complex real integer ext-real ) set ) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & (((m1 : ( ( integer ) ( complex real integer ext-real ) Integer) div (a : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m1 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) : ( ( integer ) ( complex real integer ext-real ) set ) * w : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - (s : ( ( integer ) ( complex real integer ext-real ) Integer) - r : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) ) : ( ( ) ( complex real integer ext-real ) set ) mod (m2 : ( ( integer ) ( complex real integer ext-real ) Integer) div (b : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m2 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) ) : ( ( integer ) ( complex real integer ext-real ) set ) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) ;

theorem :: INT_4:44
for m1, m2, m3, a, b, c being ( ( integer ) ( complex real integer ext-real ) Integer) st m1 : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & m2 : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & m3 : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & m1 : ( ( integer ) ( complex real integer ext-real ) Integer) ,m2 : ( ( integer ) ( complex real integer ext-real ) Integer) are_relative_prime & m1 : ( ( integer ) ( complex real integer ext-real ) Integer) ,m3 : ( ( integer ) ( complex real integer ext-real ) Integer) are_relative_prime & m2 : ( ( integer ) ( complex real integer ext-real ) Integer) ,m3 : ( ( integer ) ( complex real integer ext-real ) Integer) are_relative_prime holds
ex r, s being ( ( integer ) ( complex real integer ext-real ) Integer) st
for x being ( ( integer ) ( complex real integer ext-real ) Integer) st (x : ( ( integer ) ( complex real integer ext-real ) Integer) - a : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m1 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & (x : ( ( integer ) ( complex real integer ext-real ) Integer) - b : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m2 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & (x : ( ( integer ) ( complex real integer ext-real ) Integer) - c : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m3 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
( x : ( ( integer ) ( complex real integer ext-real ) Integer) ,(a : ( ( integer ) ( complex real integer ext-real ) Integer) + (m1 : ( ( integer ) ( complex real integer ext-real ) Integer) * r : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) ) : ( ( ) ( complex real integer ext-real ) set ) + ((m1 : ( ( integer ) ( complex real integer ext-real ) Integer) * m2 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) * s : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) : ( ( ) ( complex real integer ext-real ) set ) are_congruent_mod (m1 : ( ( integer ) ( complex real integer ext-real ) Integer) * m2 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) * m3 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( complex real integer ext-real ) set ) & ((m1 : ( ( integer ) ( complex real integer ext-real ) Integer) * r : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - (b : ( ( integer ) ( complex real integer ext-real ) Integer) - a : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) ) : ( ( ) ( complex real integer ext-real ) set ) mod m2 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & (((m1 : ( ( integer ) ( complex real integer ext-real ) Integer) * m2 : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) * s : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - ((c : ( ( integer ) ( complex real integer ext-real ) Integer) - a : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - (m1 : ( ( integer ) ( complex real integer ext-real ) Integer) * r : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) ) : ( ( ) ( complex real integer ext-real ) set ) ) : ( ( ) ( complex real integer ext-real ) set ) mod m3 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) ;

theorem :: INT_4:45
for m1, m2, m3, a, b, c being ( ( integer ) ( complex real integer ext-real ) Integer) st m1 : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & m2 : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & m3 : ( ( integer ) ( complex real integer ext-real ) Integer) <> 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & ( not m1 : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m2 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) divides a : ( ( integer ) ( complex real integer ext-real ) Integer) - b : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( complex real integer ext-real ) set ) or not m1 : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m3 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) divides a : ( ( integer ) ( complex real integer ext-real ) Integer) - c : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( complex real integer ext-real ) set ) or not m2 : ( ( integer ) ( complex real integer ext-real ) Integer) gcd m3 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) set ) divides b : ( ( integer ) ( complex real integer ext-real ) Integer) - c : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( complex real integer ext-real ) set ) ) holds
for x being ( ( integer ) ( complex real integer ext-real ) Integer) holds
( not (x : ( ( integer ) ( complex real integer ext-real ) Integer) - a : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m1 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) or not (x : ( ( integer ) ( complex real integer ext-real ) Integer) - b : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m2 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) or not (x : ( ( integer ) ( complex real integer ext-real ) Integer) - c : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod m3 : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) ;

theorem :: INT_4:46
for n1, n2, n3 being ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) holds (n1 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) gcd n3 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) lcm (n2 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) gcd n3 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = (n1 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) lcm n2 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) gcd n3 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:47
for a, b, c being ( ( integer ) ( complex real integer ext-real ) Integer)
for n1, n2, n3 being ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) st n1 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) gcd n2 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) divides a : ( ( integer ) ( complex real integer ext-real ) Integer) - b : ( ( integer ) ( complex real integer ext-real ) Integer) : ( ( ) ( complex real integer ext-real ) set ) holds
ex r, s being ( ( integer ) ( complex real integer ext-real ) Integer) st
for x being ( ( integer ) ( complex real integer ext-real ) Integer) st (x : ( ( integer ) ( complex real integer ext-real ) Integer) - a : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod n1 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & (x : ( ( integer ) ( complex real integer ext-real ) Integer) - b : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod n2 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & (x : ( ( integer ) ( complex real integer ext-real ) Integer) - c : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) mod n3 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
( x : ( ( integer ) ( complex real integer ext-real ) Integer) ,(a : ( ( integer ) ( complex real integer ext-real ) Integer) + (n1 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) * r : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) ) : ( ( ) ( complex real integer ext-real ) set ) + ((n1 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) lcm n2 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) * s : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) : ( ( ) ( complex real integer ext-real ) set ) are_congruent_mod (n1 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) lcm n2 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) lcm n3 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & (((n1 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) div (n1 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) gcd n2 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) * r : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - ((b : ( ( integer ) ( complex real integer ext-real ) Integer) - a : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) div (n1 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) gcd n2 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( integer ) ( complex real integer ext-real ) set ) ) : ( ( ) ( complex real integer ext-real ) set ) mod (n2 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) div (n1 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) gcd n2 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & ((((n1 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) lcm n2 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) div ((n1 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) lcm n2 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) gcd n3 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) * s : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) - ((c : ( ( integer ) ( complex real integer ext-real ) Integer) - (a : ( ( integer ) ( complex real integer ext-real ) Integer) + (n1 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) * r : ( ( integer ) ( complex real integer ext-real ) Integer) ) : ( ( ) ( complex real integer ext-real ) set ) ) : ( ( ) ( complex real integer ext-real ) set ) ) : ( ( ) ( complex real integer ext-real ) set ) div ((n1 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) lcm n2 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) gcd n3 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( integer ) ( complex real integer ext-real ) set ) ) : ( ( ) ( complex real integer ext-real ) set ) mod (n3 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) div ((n1 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) lcm n2 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) gcd n3 : ( ( non zero natural ) ( non zero epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real positive non negative ) Nat) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( integer ) ( complex real integer ext-real ) set ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) ;

begin

definition
let m be ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) ;
let X be ( ( ) ( ) set ) ;
pred X is_CRS_of m means :: INT_4:def 2
 ex fp being ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() ) FinSequence of INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) ) st
( X : ( ( ) ( ) set ) = rng fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() ) FinSequence of INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) & len fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() ) FinSequence of INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = m : ( ( ) ( ) set ) & ( for b being ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) st b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) in dom fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() ) FinSequence of INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) holds
fp : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued Function-like finite FinSequence-like FinSubsequence-like V71() V72() V73() ) FinSequence of INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) ) . b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) : ( ( ) ( complex real integer ext-real ) set ) in Class ((Cong m : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued V71() V72() V73() ) Relation of ) ,(b : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer finite cardinal ext-real non negative ) Nat) -' 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real positive non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) : ( ( ) ( complex-membered ext-real-membered real-membered rational-membered integer-membered ) Element of K6(INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) );
end;

theorem :: INT_4:48
for m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds { a : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) where a is ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : a : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) < m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) } is_CRS_of m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:49
for m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) )
for X being ( ( finite ) ( finite ) set ) st X : ( ( finite ) ( finite ) set ) is_CRS_of m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
( card X : ( ( finite ) ( finite ) set ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) set ) ) = m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & ( for x, y being ( ( integer ) ( complex real integer ext-real ) Integer) st x : ( ( integer ) ( complex real integer ext-real ) Integer) in X : ( ( finite ) ( finite ) set ) & y : ( ( integer ) ( complex real integer ext-real ) Integer) in X : ( ( finite ) ( finite ) set ) & x : ( ( integer ) ( complex real integer ext-real ) Integer) <> y : ( ( integer ) ( complex real integer ext-real ) Integer) holds
not [x : ( ( integer ) ( complex real integer ext-real ) Integer) ,y : ( ( integer ) ( complex real integer ext-real ) Integer) ] : ( ( ) ( ) set ) in Cong m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( Relation-like INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued total reflexive symmetric transitive V71() V72() V73() ) Relation of ) ) ) ;

theorem :: INT_4:50
for m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
( {} : ( ( ) ( ) set ) is_CRS_of m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) iff m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = 0 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ) ;

theorem :: INT_4:51
for m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) )
for X being ( ( finite ) ( finite ) set ) st card X : ( ( finite ) ( finite ) set ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) set ) ) = m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
ex fp being ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) st
( len fp : ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) = m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & ( for a being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) st a : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) in dom fp : ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) : ( ( ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of K6(NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( V2() non finite ) set ) ) holds
fp : ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) . a : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( ) set ) in X : ( ( finite ) ( finite ) set ) ) & fp : ( ( Relation-like Function-like FinSequence-like ) ( Relation-like NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) -defined Function-like finite FinSequence-like FinSubsequence-like ) FinSequence) is one-to-one ) ;

theorem :: INT_4:52
for m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) )
for X being ( ( finite ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered ) Subset of ( ( ) ( V2() non finite ) set ) ) st card X : ( ( finite ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered ) Subset of ( ( ) ( V2() non finite ) set ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) set ) ) = m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) & ( for x, y being ( ( integer ) ( complex real integer ext-real ) Integer) st x : ( ( integer ) ( complex real integer ext-real ) Integer) in X : ( ( finite ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered ) Subset of ( ( ) ( V2() non finite ) set ) ) & y : ( ( integer ) ( complex real integer ext-real ) Integer) in X : ( ( finite ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered ) Subset of ( ( ) ( V2() non finite ) set ) ) & x : ( ( integer ) ( complex real integer ext-real ) Integer) <> y : ( ( integer ) ( complex real integer ext-real ) Integer) holds
not [x : ( ( integer ) ( complex real integer ext-real ) Integer) ,y : ( ( integer ) ( complex real integer ext-real ) Integer) ] : ( ( ) ( ) set ) in Cong m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) : ( ( ) ( Relation-like INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -defined INT : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V87() ) set ) -valued total reflexive symmetric transitive V71() V72() V73() ) Relation of ) ) holds
X : ( ( finite ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered ) Subset of ( ( ) ( V2() non finite ) set ) ) is_CRS_of m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:53
for m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) )
for a being ( ( integer ) ( complex real integer ext-real ) Integer)
for X being ( ( finite ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered ) Subset of ( ( ) ( V2() non finite ) set ) ) st X : ( ( finite ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered ) Subset of ( ( ) ( V2() non finite ) set ) ) is_CRS_of m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
a : ( ( integer ) ( complex real integer ext-real ) Integer) ++ X : ( ( finite ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered ) Subset of ( ( ) ( V2() non finite ) set ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) is_CRS_of m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;

theorem :: INT_4:54
for m being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) )
for a being ( ( integer ) ( complex real integer ext-real ) Integer)
for X being ( ( finite ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered ) Subset of ( ( ) ( V2() non finite ) set ) ) st a : ( ( integer ) ( complex real integer ext-real ) Integer) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) are_relative_prime & X : ( ( finite ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered ) Subset of ( ( ) ( V2() non finite ) set ) ) is_CRS_of m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) holds
a : ( ( integer ) ( complex real integer ext-real ) Integer) ** X : ( ( finite ) ( finite complex-membered ext-real-membered real-membered rational-membered integer-membered ) Subset of ( ( ) ( V2() non finite ) set ) ) : ( ( ) ( finite complex-membered ext-real-membered real-membered ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) is_CRS_of m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural complex real integer V31() finite cardinal ext-real non negative complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered ) Element of NAT : ( ( ) ( non empty V2() epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V87() ) Element of K6(REAL : ( ( ) ( non empty V2() non finite complex-membered ext-real-membered real-membered V87() ) set ) ) : ( ( ) ( V2() non finite ) set ) ) ) ;