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 Th12:
  r(#)(f1(#)f2)=r(#)f1(#)f2
proof
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;
  now
    let c be object;
    assume
A2: c in dom (r(#)(f1(#)f2));
    then c in dom (r(#)f1) /\ dom f2 by A1,VALUED_1:def 4;
    then
A3: 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 VALUED_1:5
      .= (r*f1.c) * f2.c
      .= (r(#)f1).c * f2.c by A3,VALUED_1:def 5
      .= (r(#)f1 (#) f2).c by VALUED_1:5;
  end;
  hence thesis by A1,FUNCT_1:2;
end;
