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
  f [+] c1 [-] c2 = f [+] (c1-c2)
proof
  set f1 = f[+]c1;
A1: dom(f1[-]c2) = dom f1 by Def37;
  dom f1 = dom f by Def37;
  hence
A2: dom(f1[-]c2) = dom(f[+](c1-c2)) by A1,Def37;
  let x be object;
  assume
A3: x in dom(f1[-]c2);
  hence (f1[-]c2).x = f1.x - c2 by Def37
    .= f.x + c1 - c2 by A1,A3,Def37
    .= f.x + (c1 - c2) by Th12
    .= (f[+](c1-c2)).x by A2,A3,Def37;
end;
