reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem Th17:
  for f1,f2 being complex-valued Function holds r(#)(f1(#)f2) = r(#)f1(#)f2
proof
  let f1,f2 be complex-valued Function;
  thus
A1: dom (r(#)(f1 (#) f2)) = dom (f1 (#) f2) by VALUED_1:def 5
    .= dom f1 /\ dom f2 by VALUED_1:def 4
    .= dom (r(#)f1) /\ dom f2 by VALUED_1:def 5
    .= dom (r(#)f1(#)f2) by VALUED_1:def 4;
    let c be object;
    assume
A2: c in dom (r(#)(f1(#)f2));
    then
A3: c in dom (f1(#)f2) by VALUED_1:def 5;
    c in dom (r(#)f1) /\ dom f2 by A1,A2,VALUED_1:def 4;
    then
A4: c in dom (r(#)f1) by 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:def 4
      .= (r*((f1.c))) * ((f2.c))
      .= (r(#)f1).c * ((f2.c)) by A4,VALUED_1:def 5
      .= (r(#)f1 (#) f2).c by A1,A2,VALUED_1:def 4;
end;
