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

proof

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;

A1 c= A2

proof

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;

hence A1 = A2 by A7; :: thesis: verum