let IT1, IT2 be Subset of (the_Edges_of G); :: thesis: ( ( for e being set holds
( e in IT1 iff ( e in the_Edges_of G & (the_Target_of G) . e in X ) ) ) & ( for e being set holds
( e in IT2 iff ( e in the_Edges_of G & (the_Target_of G) . e in X ) ) ) implies IT1 = IT2 )
assume that
A2: for e being set holds
( e in IT1 iff ( e in the_Edges_of G & (the_Target_of G) . e in X ) ) and
A3: for e being set holds
( e in IT2 iff ( e in the_Edges_of G & (the_Target_of G) . e in X ) ) ; :: thesis: IT1 = IT2

now :: thesis: for e being object holds
( ( e in IT1 implies e in IT2 ) & ( e in IT2 implies e in IT1 ) )

let e be object ; :: thesis: ( ( e in IT1 implies e in IT2 ) & ( e in IT2 implies e in IT1 ) )

hereby :: thesis: ( e in IT2 implies e in IT1 )

assume A4: e in IT1 ; :: thesis: e in IT2
then (the_Target_of G) . e in X by A2;
hence e in IT2 by A3, A4; :: thesis: verum

end;

assume A5: e in IT2 ; :: thesis: e in IT1
then (the_Target_of G) . e in X by A3;
hence e in IT1 by A2, A5; :: thesis: verum

end;

hence IT1 = IT2 by TARSKI:2; :: thesis: verum