let C be non empty set ; :: thesis: for V being RealNormSpace
for f2 being PartFunc of C,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,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,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 object ; :: 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) by XBOOLE_0:def 5;
reconsider x1 = x as Element of C by A1;
not x in (f1 (#) f2) " {(0. V)} by A1, XBOOLE_0:def 5;
then not (f1 (#) f2) /. x1 in {(0. V)} by A2, PARTFUN2:26;
then (f1 (#) f2) /. x1 <> 0. V by TARSKI:def 1;
then A3: (f1 . x1) * (f2 /. x1) <> 0. V by A2, Def3;
then f2 /. x1 <> 0. V ;
then not f2 /. x1 in {(0. V)} by TARSKI:def 1;
then A4: not x1 in f2 " {(0. V)} by PARTFUN2:26;
A5: x1 in (dom f1) /\ (dom f2) by A2, Def3;
then x1 in dom f2 by XBOOLE_0:def 4;
then A6: x in (dom f2) \ (f2 " {(0. V)}) by A4, XBOOLE_0:def 5;
f1 . x1 <> 0 by A3, RLVECT_1:10;
then not f1 . x1 in {0} by TARSKI:def 1;
then A7: not x1 in f1 " {0} by FUNCT_1:def 7;
x1 in dom f1 by A5, XBOOLE_0:def 4;
then x in (dom f1) \ (f1 " {0}) by A7, XBOOLE_0:def 5;
hence x in ((dom f1) \ (f1 " {0})) /\ ((dom f2) \ (f2 " {(0. V)})) by A6, 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 object ; :: 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 A8: x in ((dom f1) \ (f1 " {0})) /\ ((dom f2) \ (f2 " {(0. V)})) ; :: thesis: x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " {(0. V)})
then reconsider x1 = x as Element of C ;
A9: x in (dom f2) \ (f2 " {(0. V)}) by A8, XBOOLE_0:def 4;
then A10: x in dom f2 by XBOOLE_0:def 5;
not x in f2 " {(0. V)} by A9, XBOOLE_0:def 5;
then not f2 /. x1 in {(0. V)} by A10, PARTFUN2:26;
then A11: f2 /. x1 <> 0. V by TARSKI:def 1;
A12: x in (dom f1) \ (f1 " {0}) by A8, XBOOLE_0:def 4;
then A13: x in dom f1 by XBOOLE_0:def 5;
then x1 in (dom f1) /\ (dom f2) by A10, XBOOLE_0:def 4;
then A14: x1 in dom (f1 (#) f2) by Def3;
not x in f1 " {0} by A12, XBOOLE_0:def 5;
then not f1 . x1 in {0} by A13, FUNCT_1:def 7;
then f1 . x1 <> 0 by TARSKI:def 1;
then (f1 . x1) * (f2 /. x1) <> 0. V by A11, RLVECT_1:11;
then (f1 (#) f2) /. x1 <> 0. V by A14, Def3;
then not (f1 (#) f2) /. x1 in {(0. V)} by TARSKI:def 1;
then not x in (f1 (#) f2) " {(0. V)} by PARTFUN2:26;
hence x in (dom (f1 (#) f2)) \ ((f1 (#) f2) " {(0. V)}) by A14, XBOOLE_0:def 5; :: thesis: verum

end;