let X1, X2 be set ; :: thesis: ( ( for V being object holds
( V in X1 iff ex G being _Graph st
( G in S & V = the_Vertices_of G ) ) ) & ( for V being object holds
( V in X2 iff ex G being _Graph st
( G in S & V = the_Vertices_of G ) ) ) implies X1 = X2 )
assume that
A3: for V being object holds
( V in X1 iff ex G being _Graph st
( G in S & V = the_Vertices_of G ) ) and
A4: for V being object holds
( V in X2 iff ex G being _Graph st
( G in S & V = the_Vertices_of G ) ) ; :: thesis: X1 = X2

now :: thesis: for V being object holds
( ( V in X1 implies V in X2 ) & ( V in X2 implies V in X1 ) )

let V be object ; :: thesis: ( ( V in X1 implies V in X2 ) & ( V in X2 implies V in X1 ) )

hereby :: thesis: ( V in X2 implies V in X1 )

assume V in X1 ; :: thesis: V in X2
then ex G being _Graph st
( G in S & V = the_Vertices_of G ) by A3;
hence V in X2 by A4; :: thesis: verum

end;

assume V in X2 ; :: thesis: V in X1
then ex G being _Graph st
( G in S & V = the_Vertices_of G ) by A4;
hence V in X1 by A3; :: thesis: verum

end;

hence X1 = X2 by TARSKI:2; :: thesis: verum