reserve x, y for object, X, X1, X2 for set;
reserve Y, Y1, Y2 for complex-functions-membered set,
  c, c1, c2 for Complex,
  f for PartFunc of X,Y,
  f1 for PartFunc of X1,Y1,
  f2 for PartFunc of X2, Y2,
  g, h, k for complex-valued Function;

theorem
  abs abs f = abs f
proof
  set f1 = abs f;
  thus
A1: dom abs f1 = dom abs f by Def36;
  let x be object;
  assume
A2: x in dom abs f1;
  hence (abs f1).x = abs(f1.x) by Def36
    .= abs(abs(f.x)) by A1,A2,Def36
    .= (abs f).x by A1,A2,Def36;
end;
