let F1, F2 be MSFunctionSet of U1,U1; :: thesis: ( ( for h being ManySortedFunction of U1,U1 holds
( h in F1 iff h is_isomorphism U1,U1 ) ) & ( for h being ManySortedFunction of U1,U1 holds
( h in F2 iff h is_isomorphism U1,U1 ) ) implies F1 = F2 )
assume that
A3: for h being ManySortedFunction of U1,U1 holds
( h in F1 iff h is_isomorphism U1,U1 ) and
A4: for h being ManySortedFunction of U1,U1 holds
( h in F2 iff h is_isomorphism U1,U1 ) ; :: thesis: F1 = F2
thus F1 c= F2 :: according to XBOOLE_0:def 10 :: thesis: F2 c= F1
proof
let q be object ; :: according to TARSKI:def 3 :: thesis: ( not q in F1 or q in F2 )
assume A5: q in F1 ; :: thesis: q in F2
then reconsider h1 = q as ManySortedFunction of U1,U1 by Th19;
h1 is_isomorphism U1,U1 by A3, A5;
hence q in F2 by A4; :: thesis: verum
end;
let q be object ; :: according to TARSKI:def 3 :: thesis: ( not q in F2 or q in F1 )
assume A6: q in F2 ; :: thesis: q in F1
then reconsider h1 = q as ManySortedFunction of U1,U1 by Th19;
h1 is_isomorphism U1,U1 by A4, A6;
hence q in F1 by A3; :: thesis: verum