let A1, A2 be set ; :: thesis: ( ( for x being set holds

( x in A1 iff ex f, g being Function st

( x = [f,g] & f,g form_morphism_between S1,S2 ) ) ) & ( for x being set holds

( x in A2 iff ex f, g being Function st

( x = [f,g] & f,g form_morphism_between S1,S2 ) ) ) implies A1 = A2 )

assume that

A5: for x being set holds

( x in A1 iff ex f, g being Function st

( x = [f,g] & f,g form_morphism_between S1,S2 ) ) and

A6: for x being set holds

( x in A2 iff ex f, g being Function st

( x = [f,g] & f,g form_morphism_between S1,S2 ) ) ; :: thesis: A1 = A2

A7: A2 c= A1

( x in A1 iff ex f, g being Function st

( x = [f,g] & f,g form_morphism_between S1,S2 ) ) ) & ( for x being set holds

( x in A2 iff ex f, g being Function st

( x = [f,g] & f,g form_morphism_between S1,S2 ) ) ) implies A1 = A2 )

assume that

A5: for x being set holds

( x in A1 iff ex f, g being Function st

( x = [f,g] & f,g form_morphism_between S1,S2 ) ) and

A6: for x being set holds

( x in A2 iff ex f, g being Function st

( x = [f,g] & f,g form_morphism_between S1,S2 ) ) ; :: thesis: A1 = A2

A7: A2 c= A1

proof

A1 c= A2
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in A2 or x in A1 )

assume x in A2 ; :: thesis: x in A1

then ex f, g being Function st

( x = [f,g] & f,g form_morphism_between S1,S2 ) by A6;

hence x in A1 by A5; :: thesis: verum

end;assume x in A2 ; :: thesis: x in A1

then ex f, g being Function st

( x = [f,g] & f,g form_morphism_between S1,S2 ) by A6;

hence x in A1 by A5; :: thesis: verum

proof

hence
A1 = A2
by A7; :: thesis: verum
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in A1 or x in A2 )

assume x in A1 ; :: thesis: x in A2

then ex f, g being Function st

( x = [f,g] & f,g form_morphism_between S1,S2 ) by A5;

hence x in A2 by A6; :: thesis: verum

end;assume x in A1 ; :: thesis: x in A2

then ex f, g being Function st

( x = [f,g] & f,g form_morphism_between S1,S2 ) by A5;

hence x in A2 by A6; :: thesis: verum