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 Th30:
  (abs(f))^ = abs(f^)
proof
A1: dom ((abs(f))^) = dom (abs(f)) \ (abs(f))"{0} by Def2
    .= dom f \ (abs(f))"{0} by VALUED_1:def 11
    .= dom f \ f"{0} by Th5
    .= dom (f^) by Def2
    .= dom (abs((f^))) by VALUED_1:def 11;
  now
    let c be object;
    assume
A2: c in dom ((abs(f))^);
    then
A3: c in dom (f^) by A1,VALUED_1:def 11;
    thus ((abs(f))^).c = ((abs(f)).c)" by A2,Def2
      .= (|.f.c.|)" by VALUED_1:18
      .= 1/|.f.c.| by XCMPLX_1:215
      .= |.1/f.c.| by COMPLEX1:80
      .= |.(f.c)".| by XCMPLX_1:215
      .= |.f^.c.| by A3,Def2
      .= (abs(f^)).c by VALUED_1:18;
  end;
  hence thesis by A1,FUNCT_1:2;
end;
