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
  abs(r(#)f) = |.r.|(#)abs(f)
proof
A1: dom (abs(r(#)f)) = dom (r(#)f) by VALUED_1:def 11
    .= dom f by VALUED_1:def 5
    .= dom (abs(f)) by VALUED_1:def 11
    .= dom (|.r.|(#)abs(f)) by VALUED_1:def 5;
  now
    let c be object;
    assume
A2: c in dom (abs(r(#)f));
    then
A3: c in dom (r(#)f) by VALUED_1:def 11;
    thus (abs(r(#)f)).c = |.(r(#)f).c.| by VALUED_1:18
      .=|.r*(f.c).| by A3,VALUED_1:def 5
      .=|.r.|*|.f.c.| by COMPLEX1:65
      .=|.r.|*(abs(f)).c by VALUED_1:18
      .=(|.r.|(#)abs(f)).c by A1,A2,VALUED_1:def 5;
  end;
  hence thesis by A1,FUNCT_1:2;
end;
