reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;

theorem Th14:
  (f1 - f2)(#)f3=f1(#)f3 - f2(#)f3
proof
A1: dom ((f1 - f2) (#) f3) = dom (f1 - f2) /\ dom f3 by VALUED_1:def 4
    .= dom f1 /\ dom f2 /\ (dom f3 /\ dom f3) by VALUED_1:12
    .= 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 VALUED_1:def 4
    .= dom (f1 (#) f3) /\ dom (f2 (#) f3) by VALUED_1:def 4
    .= dom (f1 (#) f3 - f2 (#) f3) by VALUED_1:12;
  now
    let c be object;
    assume
A2: c in dom ((f1 - f2)(#)f3);
    then c in dom (f1 - f2) /\ dom f3 by VALUED_1:def 4;
    then
A3: c in dom (f1 - f2) by XBOOLE_0:def 4;
    thus ((f1 - f2) (#) f3).c = (f1 - f2).c * f3.c by VALUED_1:5
      .= (f1.c - f2.c) * f3.c by A3,VALUED_1:13
      .= f1.c * f3.c - f2.c * f3.c
      .= (f1 (#) f3).c - f2.c * f3.c by VALUED_1:5
      .= (f1 (#) f3).c - (f2 (#) f3).c by VALUED_1:5
      .=((f1 (#) f3) - (f2 (#) f3)).c by A1,A2,VALUED_1:13;
  end;
  hence thesis by A1,FUNCT_1:2;
end;
