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 Th24:
  abs(f1(#)f2) = abs(f1)(#)abs(f2)
proof
A1: now
    let c be object;
    assume c in dom (abs(f1 (#) f2));
    thus (abs(f1(#)f2)).c = |.(f1(#)f2).c.| by VALUED_1:18
      .= |.f1.c * f2.c.| by VALUED_1:5
      .= |.f1.c.| * |.f2.c.| by COMPLEX1:65
      .= ((abs(f1)).c) * |.f2.c.| by VALUED_1:18
      .= ((abs(f1)).c) * (abs(f2)).c by VALUED_1:18
      .= (abs(f1)(#)abs(f2)).c by VALUED_1:5;
  end;
  dom (abs(f1 (#) f2)) = dom (f1 (#) f2) by VALUED_1:def 11
    .= dom f1 /\ dom f2 by VALUED_1:def 4
    .= dom f1 /\ dom (abs(f2)) by VALUED_1:def 11
    .= dom (abs(f1)) /\ dom (abs(f2)) by VALUED_1:def 11
    .= dom (abs(f1)(#)abs(f2)) by VALUED_1:def 4;
  hence thesis by A1,FUNCT_1:2;
end;
