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
  for f being complex-valued Function holds (r*q)(#)f = r(#)(q(#)f)
proof
  let f be complex-valued Function;
  thus
A1: dom ((r*q) (#) f) = dom f by VALUED_1:def 5
    .= dom (q(#)f) by VALUED_1:def 5
    .= dom (r(#)(q(#)f)) by VALUED_1:def 5;
    let c be object;
    assume
A2: c in dom ((r*q)(#)f);
    then
A3: c in dom (q(#)f) by A1,VALUED_1:def 5;
    thus ((r*q)(#)f).c = r*q * (f.c) by A2,VALUED_1:def 5
    .= r*(q * (f.c))
    .= r * ((q(#)f).c) by A3,VALUED_1:def 5
    .= (r(#)(q(#)f)).c by A1,A2,VALUED_1:def 5;
  end;
