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 Th30:
  for f being complex-valued Function holds |.r(#)f.| = |.r.|(#)|.f.|
proof
  let f be complex-valued Function;
A1: dom f = dom (|.f.|) by VALUED_1:def 11;
    thus
A2: dom (|.r(#)f.|) = dom (r(#)f) by VALUED_1:def 11
    .= dom f by VALUED_1:def 5
    .= dom (|.r.|(#)|.f.|) by A1,VALUED_1:def 5;
    let c be object;
    assume
A3: c in dom (|.r(#)f.|);
    then
A4: c in dom (r(#)f) by VALUED_1:def 11;
    thus (|.r(#)f.|).c = |.(r(#)f).c.| by VALUED_1:18
      .=|.r*((f.c)).| by A4,VALUED_1:def 5
      .=|.r.|*|.((f.c)).| by COMPLEX1:65
      .=|.r.|*((|.f.|).c) by VALUED_1:18
      .=(|.r.|(#)|.f.|).c by A2,A3,VALUED_1:def 5;
end;
