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,f2 be PartFunc of M,COMPLEX holds (f1 - f2) (#) f3 = f1(#)f3 - f2(#)f3
proof
  let f1,f2 be PartFunc of M,COMPLEX;
A1: dom ((f1 - f2) (#) f3) = dom (f1 + -f2) /\ dom f3 by Def1
    .= dom f1 /\ dom (-f2) /\ (dom f3 /\ dom f3) by VALUED_1:def 1
    .= dom f1 /\ dom f2 /\ (dom f3 /\ dom f3) by VALUED_1:8
    .= dom f1 /\ dom f2 /\ dom f3 /\ dom f3 by XBOOLE_1:16
    .= dom f1 /\ dom f3 /\ dom f2 /\ dom f3 by XBOOLE_1:16
    .= dom f1 /\ dom f3 /\ (dom f2 /\ dom f3) by XBOOLE_1:16
    .= dom (f1 (#) f3) /\ (dom f2 /\ dom f3) by Def1
    .= dom (f1 (#) f3) /\ dom (f2 (#) f3) by Def1
    .= dom (f1 (#) f3 - f2 (#) f3) by VFUNCT_1:def 2;
  now
    let x be Element of M;
    assume
A2: x in dom ((f1 - f2)(#)f3);
    then
A3: x in dom (f1(#)f3) /\ dom (f2(#)f3) by A1,VFUNCT_1:def 2;
    then
A4: x in dom (f1(#)f3) by XBOOLE_0:def 4;
    x in dom (f1 - f2) /\ dom f3 by A2,Def1;
    then
A5: x in dom (f1 - f2) by XBOOLE_0:def 4;
    then
A6: x in dom f1 /\ dom (-f2) by VALUED_1:def 1;
    then
A7: x in dom (-f2) by XBOOLE_0:def 4;
    x in dom f1 by A6,XBOOLE_0:def 4;
    then
A8: f1/.x = f1.x by PARTFUN1:def 6;
    (f1 + -f2)/.x = (f1 + -f2).x by A5,PARTFUN1:def 6;
    then
A9: (f1 + -f2)/.x = f1.x + (-f2).x by A5,VALUED_1:def 1
      .= f1/.x + (-f2)/.x by A7,A8,PARTFUN1:def 6;
    dom -f2 = dom f2 by VALUED_1:8;
    then
A10: (-f2).x = -(f2.x) & f2/.x = f2.x by A7,PARTFUN1:def 6,VALUED_1:8;
A11: x in dom (f2 (#) f3) by A3,XBOOLE_0:def 4;
    (-f2)/.x = (-f2).x by A7,PARTFUN1:def 6;
    hence ((f1 - f2) (#) f3)/.x = (f1/.x - f2/.x) * (f3/.x) by A2,A10,A9,Def1
      .= f1/.x * (f3/.x) - f2/.x * (f3/.x) by CLVECT_1:10
      .= ((f1 (#) f3)/.x) - f2/.x * (f3/.x) by A4,Def1
      .= ((f1 (#) f3)/.x) - ((f2 (#) f3)/.x) by A11,Def1
      .= ((f1 (#) f3) - (f2 (#) f3))/.x by A1,A2,VFUNCT_1:def 2;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
