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

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " {(0. V)}) or x in ((dom f1) \ (f1 " {0 })) /\ ((dom f2) \ (f2 " {(0. V)})) )
assume A1: x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " {(0. V)}) ; :: thesis: x in ((dom f1) \ (f1 " {0 })) /\ ((dom f2) \ (f2 " {(0. V)}))
then A2: ( x in dom (f1 (#) f2) & not x in (f1 (#) f2) " {(0. V)} ) by XBOOLE_0:def 5;
reconsider x1 = x as Element of C by A1;
not (f1 (#) f2) /. x1 in {(0. V)} by A2, PARTFUN2:44;
then (f1 (#) f2) /. x1 <> 0. V by TARSKI:def 1;
then (f1 . x1) * (f2 /. x1) <> 0. V by A2, Def3;
then ( f1 . x1 <> 0 & f2 /. x1 <> 0. V ) by RLVECT_1:23;
then ( x1 in (dom f1) /\ (dom f2) & not f1 . x1 in {0 } & not f2 /. x1 in {(0. V)} ) by A2, Def3, TARSKI:def 1;
then ( x1 in dom f1 & x1 in dom f2 & not x1 in f1 " {0 } & not f2 /. x1 in {(0. V)} ) by FUNCT_1:def 13, XBOOLE_0:def 4;
then ( x in (dom f1) \ (f1 " {0 }) & x1 in dom f2 & not x1 in f2 " {(0. V)} ) by PARTFUN2:44, XBOOLE_0:def 5;
then ( x in (dom f1) \ (f1 " {0 }) & x in (dom f2) \ (f2 " {(0. V)}) ) by XBOOLE_0:def 5;
hence x in ((dom f1) \ (f1 " {0 })) /\ ((dom f2) \ (f2 " {(0. V)})) by XBOOLE_0:def 4; :: thesis: verum

end;

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

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in ((dom f1) \ (f1 " {0 })) /\ ((dom f2) \ (f2 " {(0. V)})) or x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " {(0. V)}) )
assume A3: x in ((dom f1) \ (f1 " {0 })) /\ ((dom f2) \ (f2 " {(0. V)})) ; :: thesis: x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " {(0. V)})
then ( x in (dom f1) \ (f1 " {0 }) & x in (dom f2) \ (f2 " {(0. V)}) ) by XBOOLE_0:def 4;
then A4: ( x in dom f1 & not x in f1 " {0 } & x in dom f2 & not x in f2 " {(0. V)} ) by XBOOLE_0:def 5;
reconsider x1 = x as Element of C by A3;
not f1 . x1 in {0 } by A4, FUNCT_1:def 13;
then A5: f1 . x1 <> 0 by TARSKI:def 1;
not f2 /. x1 in {(0. V)} by A4, PARTFUN2:44;
then f2 /. x1 <> 0. V by TARSKI:def 1;
then A6: (f1 . x1) * (f2 /. x1) <> 0. V by A5, RLVECT_1:24;
x1 in (dom f1) /\ (dom f2) by A4, XBOOLE_0:def 4;
then A7: x1 in dom (f1 (#) f2) by Def3;
then (f1 (#) f2) /. x1 <> 0. V by A6, Def3;
then not (f1 (#) f2) /. x1 in {(0. V)} by TARSKI:def 1;
then not x in (f1 (#) f2) " {(0. V)} by PARTFUN2:44;
hence x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " {(0. V)}) by A7, XBOOLE_0:def 5; :: thesis: verum

end;