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;
