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
  r(#)(f1 - f2) = r(#)f1 - r(#)f2
proof
A1: dom (r(#)(f1 - f2)) = dom (f1 - f2) by VALUED_1:def 5
    .= dom f1 /\ dom f2 by VALUED_1:12
    .= dom f1 /\ dom (r(#)f2) by VALUED_1:def 5
    .= dom (r(#)f1) /\ dom (r(#)f2) by VALUED_1:def 5
    .= dom (r(#)f1 - r(#)f2) by VALUED_1:12;
  now
    let c be object;
    assume
A2: c in dom (r(#)(f1 - f2));
    then
A3: c in dom (f1 - f2) by VALUED_1:def 5;
A4: c in dom (r(#)f1) /\ dom (r(#)f2) by A1,A2,VALUED_1:12;
    then
A5: c in dom (r(#)f1) by XBOOLE_0:def 4;
A6: c in dom (r(#)f2) by A4,XBOOLE_0:def 4;
    thus (r(#)(f1 - f2)).c = r * (f1 - f2).c by A2,VALUED_1:def 5
      .= r * (f1.c - f2.c) by A3,VALUED_1:13
      .= r * f1.c - r * f2.c
      .= (r(#)f1).c - r * f2.c by A5,VALUED_1:def 5
      .= (r(#)f1).c - (r(#)f2).c by A6,VALUED_1:def 5
      .= (r(#)f1 - r(#)f2).c by A1,A2,VALUED_1:13;
  end;
  hence thesis by A1,FUNCT_1:2;
end;
