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
  <->f1 = <->f2 implies f1 = f2
proof
A1: dom <->f1 = dom f1 by Def33;
  assume
A2: <->f1 = <->f2;
  hence dom f1 = dom f2 by A1,Def33;
  let x be object;
  assume
A3: x in dom f1;
  thus f1.x = --f1.x .= -(<->f1).x by A1,A3,Def33
    .= --f2.x by A2,A1,A3,Def33
    .= f2.x;
end;
