let C be non empty set ; :: thesis: for f1, f2 being PartFunc of C,COMPLEX holds (dom (f1 (#) f2)) \ ((f1 (#) f2) " {0 }) = ((dom f1) \ (f1 " {0 })) /\ ((dom f2) \ (f2 " {0 }))
let f1, f2 be PartFunc of C,COMPLEX ; :: thesis: (dom (f1 (#) f2)) \ ((f1 (#) f2) " {0 }) = ((dom f1) \ (f1 " {0 })) /\ ((dom f2) \ (f2 " {0 }))
thus (dom (f1 (#) f2)) \ ((f1 (#) f2) " {0 }) c= ((dom f1) \ (f1 " {0 })) /\ ((dom f2) \ (f2 " {0 })) :: according to XBOOLE_0:def 10 :: thesis: ((dom f1) \ (f1 " {0 })) /\ ((dom f2) \ (f2 " {0 })) c= (dom (f1 (#) f2)) \ ((f1 (#) f2) " {0 })

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " {0 }) or x in ((dom f1) \ (f1 " {0 })) /\ ((dom f2) \ (f2 " {0 })) )
assume A1: x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " {0 }) ; :: thesis: x in ((dom f1) \ (f1 " {0 })) /\ ((dom f2) \ (f2 " {0 }))
then A2: ( x in dom (f1 (#) f2) & not x in (f1 (#) f2) " {0c } ) by XBOOLE_0:def 5;
reconsider x1 = x as Element of C by A1;
not (f1 (#) f2) /. x1 in {0c } by A2, PARTFUN2:44;
then (f1 (#) f2) /. x1 <> 0c by TARSKI:def 1;
then (f1 /. x1) * (f2 /. x1) <> 0c by A2, Th5;
then ( f1 /. x1 <> 0c & f2 /. x1 <> 0c ) ;
then ( x1 in (dom f1) /\ (dom f2) & not f1 /. x1 in {0c } & not f2 /. x1 in {0c } ) by A2, Th5, TARSKI:def 1;
then ( x1 in dom f1 & x1 in dom f2 & not x1 in f1 " {0c } & not f2 /. x1 in {0c } ) by PARTFUN2:44, XBOOLE_0:def 4;
then ( x in (dom f1) \ (f1 " {0c }) & x1 in dom f2 & not x1 in f2 " {0c } ) by PARTFUN2:44, XBOOLE_0:def 5;
then ( x in (dom f1) \ (f1 " {0c }) & x in (dom f2) \ (f2 " {0c }) ) by XBOOLE_0:def 5;
hence x in ((dom f1) \ (f1 " {0 })) /\ ((dom f2) \ (f2 " {0 })) by XBOOLE_0:def 4; :: thesis: verum

end;

thus ((dom f1) \ (f1 " {0 })) /\ ((dom f2) \ (f2 " {0 })) c= (dom (f1 (#) f2)) \ ((f1 (#) f2) " {0 }) :: thesis: verum

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in ((dom f1) \ (f1 " {0 })) /\ ((dom f2) \ (f2 " {0 })) or x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " {0 }) )
assume A3: x in ((dom f1) \ (f1 " {0 })) /\ ((dom f2) \ (f2 " {0 })) ; :: thesis: x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " {0 })
then ( x in (dom f1) \ (f1 " {0c }) & x in (dom f2) \ (f2 " {0c }) ) by XBOOLE_0:def 4;
then A4: ( x in dom f1 & not x in f1 " {0c } & x in dom f2 & not x in f2 " {0c } ) by XBOOLE_0:def 5;
reconsider x1 = x as Element of C by A3;
not f1 /. x1 in {0c } by A4, PARTFUN2:44;
then A5: f1 /. x1 <> 0c by TARSKI:def 1;
not f2 /. x1 in {0c } by A4, PARTFUN2:44;
then f2 /. x1 <> 0c by TARSKI:def 1;
then A6: (f1 /. x1) * (f2 /. x1) <> 0c by A5, XCMPLX_1:6;
x1 in (dom f1) /\ (dom f2) by A4, XBOOLE_0:def 4;
then A7: x1 in dom (f1 (#) f2) by Th5;
then (f1 (#) f2) /. x1 <> 0c by A6, Th5;
then not (f1 (#) f2) /. x1 in {0c } by TARSKI:def 1;
then not x in (f1 (#) f2) " {0c } by PARTFUN2:44;
hence x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " {0 }) by A7, XBOOLE_0:def 5; :: thesis: verum

end;