reserve x,y,z for Element of REAL+;

theorem
  not z <=' y implies x - (z -' y) = x + y - z
proof
  assume
A1: not z <=' y;
  per cases;
  suppose
A2: z -' y <=' x;
    then z <=' x + y by Lm7;
    then
A3: x + y - z = x + y -' z by Def2;
    x - (z -' y) = x -' (z -' y) by A2,Def2;
    hence thesis by A1,A3,Lm11;
  end;
  suppose
A4: not z -' y <=' x;
    then
A5: not z <=' x + y by Lm7;
    (z -' y) -' x = z -' (x + y) by Lm9;
    hence x - (z -' y) = [{},z -' (x + y)] by A4,Def2
      .= x + y - z by A5,Def2;
  end;
end;
