reserve M for non empty set;
reserve V for ComplexNormSpace;
reserve f,f1,f2,f3 for PartFunc of M,V;
reserve z,z1,z2 for Complex;

theorem
  for f1 be PartFunc of M,COMPLEX, f2 be PartFunc of M,V
holds dom (f1(#)f2) \ (f1(#)f2)"{0.V} = (dom f1 \ (f1)"{0}) /\ (dom f2 \ (f2)"{
  0.V})
proof
  let f1 be PartFunc of M,COMPLEX;
  let f2 be PartFunc of M,V;
  thus dom (f1(#)f2) \ (f1(#)f2)"{0.V} c= (dom f1 \ (f1)"{0}) /\ (dom f2 \ (f2
  )"{0.V})
  proof
    let x be object;
    assume
A1: x in dom (f1(#)f2) \ (f1(#)f2)"{0.V};
    then reconsider x1=x as Element of M;
A2: x in dom (f1(#)f2) by A1,XBOOLE_0:def 5;
    then
A3: x1 in dom f1 /\ dom f2 by Def1;
    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
A4: (f1/.x1) * (f2/.x1) <> 0.V by A2,Def1;
    then (f1/.x1) <> 0c by CLVECT_1:1;
    then
A5: not f1/.x1 in {0} by TARSKI:def 1;
    (f2/.x1) <> 0.V by A4,CLVECT_1:1;
    then not (f2/.x1) in {0.V} by TARSKI:def 1;
    then
A6: not x1 in (f2)"{0.V} by PARTFUN2:26;
    x1 in dom f2 by A3,XBOOLE_0:def 4;
    then
A7: x in dom f2 \ (f2)"{0.V} by A6,XBOOLE_0:def 5;
    x1 in dom f1 by A3,XBOOLE_0:def 4;
    then not f1.x1 in {0} by A5,PARTFUN1:def 6;
    then
A8: not x1 in (f1)"{0} by FUNCT_1:def 7;
    x1 in dom f1 by A3,XBOOLE_0:def 4;
    then x in dom f1 \ (f1)"{0} by A8,XBOOLE_0:def 5;
    hence thesis by A7,XBOOLE_0:def 4;
  end;
  thus (dom f1 \ (f1)"{0}) /\ (dom f2 \ (f2)"{0.V}) c= dom (f1(#)f2) \ (f1(#)
  f2)"{0.V}
  proof
    let x be object;
    assume
A9: x in (dom f1 \ (f1)"{0}) /\ (dom f2 \ (f2)"{0.V});
    then reconsider x1=x as Element of M;
A10: x in dom f1 \ (f1)"{0} by A9,XBOOLE_0:def 4;
    then
A11: x in dom f1 by XBOOLE_0:def 5;
    not x in (f1)"{0} by A10,XBOOLE_0:def 5;
    then not f1.x1 in {0} by A11,FUNCT_1:def 7;
    then f1.x1 <> 0 by TARSKI:def 1;
    then
A12: f1/.x1 <> 0 by A11,PARTFUN1:def 6;
A13: x in dom f2 \ (f2)"{0.V} by A9,XBOOLE_0:def 4;
    then
A14: x in dom f2 by XBOOLE_0:def 5;
    then x1 in dom f1 /\ dom f2 by A11,XBOOLE_0:def 4;
    then
A15: x1 in dom (f1(#)f2) by Def1;
    not x in (f2)"{0.V} by A13,XBOOLE_0:def 5;
    then not (f2/.x1) in {0.V} by A14,PARTFUN2:26;
    then (f2/.x1) <> 0.V by TARSKI:def 1;
    then f1/.x1 * (f2/.x1) <>0.V by A12,CLVECT_1:2;
    then (f1(#)f2)/.x1 <> 0.V by A15,Def1;
    then not (f1(#)f2)/.x1 in {0.V} by TARSKI:def 1;
    then not x in (f1(#)f2)"{0.V} by PARTFUN2:26;
    hence thesis by A15,XBOOLE_0:def 5;
  end;
end;
