reserve x,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve V for RealNormSpace;
reserve f,f1,f2,f3 for PartFunc of C,V;
reserve r,r1,r2,p for Real;

theorem
  for f1 be PartFunc of C,REAL holds
  dom (f1(#)f2) \ (f1(#)f2)"{0.V} =
  (dom f1 \ (f1)"{0}) /\ (dom f2 \ (f2)"{0.V})
proof
  let f1 be PartFunc of C,REAL;
  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
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 thesis by A6,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
A8: x in (dom f1 \ (f1)"{0}) /\ (dom f2 \ (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 thesis by A14,XBOOLE_0:def 5;
  end;
end;
