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

theorem
  x <=' y & z <=' y implies y -' z - x = y -' x - z
proof
  assume that
A1: x <=' y and
A2: z <=' y;
  per cases;
  suppose
A3: x + z <=' y;
    then z <=' y -' x by A1,Lm8;
    then
A4: y -' x -' z = y -' x - z by Def2;
    x <=' y -' z by A2,A3,Lm8;
    then y -' z -' x = y -' z - x by Def2;
    hence thesis by A4,Lm10;
  end;
  suppose that
A5: not x + z <=' y and
A6: y <=' x;
A7: not x <=' y -' z by A2,A5,Lm8;
A8: not z <=' y -' x by A1,A5,Lm8;
A9: x = y by A1,A6,Th4;
    then x -' (x -' z) = z by A2,Lm6
      .= z -' (x -' x) by Lm3,Lm4;
    hence y -' z - x = [{},z -' (y -' x)] by A7,A9,Def2
      .= y -' x - z by A8,Def2;
  end;
  suppose that
A10: not x + z <=' y and
A11: y <=' z;
A12: not x <=' y -' z by A2,A10,Lm8;
A13: not z <=' y -' x by A1,A10,Lm8;
A14: z = y by A2,A11,Th4;
    x -' (z -' z) = x by Lm3,Lm4
      .= z -' (z -' x) by A1,A14,Lm6;
    hence y -' z - x = [{},z -' (y -' x)] by A12,A14,Def2
      .= y -' x - z by A13,Def2;
  end;
  suppose that
A15: not x + z <=' y and
A16: not y <=' x & not y <=' z;
A17: not z <=' y -' x by A1,A15,Lm8;
    ( not x <=' y -' z)& x -' (y -' z) = z -' (y -' x) by A15,A16,Lm8,Lm13;
    hence y -' z - x = [{},z -' (y -' x)] by Def2
      .= y -' x - z by A17,Def2;
  end;
end;
