let f1, f2 be complex-valued Function; :: thesis: (dom (f1 (#) f2)) \ ((f1 (#) f2) " ) = ((dom f1) \ (f1 " )) /\ ((dom f2) \ (f2 " ))
thus (dom (f1 (#) f2)) \ ((f1 (#) f2) " ) c= ((dom f1) \ (f1 " )) /\ ((dom f2) \ (f2 " )) :: according to XBOOLE_0:def 10 :: thesis: ((dom f1) \ (f1 " )) /\ ((dom f2) \ (f2 " )) c= (dom (f1 (#) f2)) \ ((f1 (#) f2) " )
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " ) or x in ((dom f1) \ (f1 " )) /\ ((dom f2) \ (f2 " )) )
assume A1: x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " ) ; :: thesis: x in ((dom f1) \ (f1 " )) /\ ((dom f2) \ (f2 " ))
then not x in (f1 (#) f2) " by XBOOLE_0:def 5;
then not (f1 (#) f2) . x in by ;
then (f1 (#) f2) . x <> 0 by TARSKI:def 1;
then A2: (f1 . x) * (f2 . x) <> 0 by VALUED_1:5;
then f2 . x <> 0 ;
then not f2 . x in by TARSKI:def 1;
then A3: not x in f2 " by FUNCT_1:def 7;
x in dom (f1 (#) f2) by A1;
then A4: x in (dom f1) /\ (dom f2) by VALUED_1:def 4;
then x in dom f2 by XBOOLE_0:def 4;
then A5: x in (dom f2) \ (f2 " ) by ;
f1 . x <> 0 by A2;
then not f1 . x in by TARSKI:def 1;
then A6: not x in f1 " by FUNCT_1:def 7;
x in dom f1 by ;
then x in (dom f1) \ (f1 " ) by ;
hence x in ((dom f1) \ (f1 " )) /\ ((dom f2) \ (f2 " )) by ; :: thesis: verum
end;
thus ((dom f1) \ (f1 " )) /\ ((dom f2) \ (f2 " )) c= (dom (f1 (#) f2)) \ ((f1 (#) f2) " ) :: thesis: verum
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in ((dom f1) \ (f1 " )) /\ ((dom f2) \ (f2 " )) or x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " ) )
assume A7: x in ((dom f1) \ (f1 " )) /\ ((dom f2) \ (f2 " )) ; :: thesis: x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " )
then x in (dom f2) \ (f2 " ) by XBOOLE_0:def 4;
then not x in f2 " by XBOOLE_0:def 5;
then not f2 . x in by ;
then A8: f2 . x <> 0 by TARSKI:def 1;
A9: x in (dom f1) \ (f1 " ) by ;
then not x in f1 " by XBOOLE_0:def 5;
then not f1 . x in by ;
then f1 . x <> 0 by TARSKI:def 1;
then (f1 . x) * (f2 . x) <> 0 by A8;
then (f1 (#) f2) . x <> 0 by VALUED_1:5;
then not (f1 (#) f2) . x in by TARSKI:def 1;
then A10: not x in (f1 (#) f2) " by FUNCT_1:def 7;
x in (dom f1) /\ (dom f2) by ;
then x in dom (f1 (#) f2) by VALUED_1:def 4;
hence x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " ) by ; :: thesis: verum
end;