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

theorem
  not y <=' x & not y <=' z implies x - (y -' z) = z - (y -' x)
proof
  assume
A1: ( not y <=' x)& not y <=' z;
  per cases;
  suppose
A2: y <=' x + z;
    then y -' x <=' z by Lm7;
    then
A3: z - (y -' x) = z -' (y -' x) by Def2;
    y -' z <=' x by A2,Lm7;
    then x - (y -' z) = x -' (y -' z) by Def2;
    hence thesis by A1,A3,Lm13;
  end;
  suppose
A4: not y <=' x + z;
    then
A5: not y -' x <=' z by Lm7;
A6: y -' z -' x = y -' x -' z by Lm10;
    not y -' z <=' x by A4,Lm7;
    hence x - (y -' z) = [{},y -' x -' z] by A6,Def2
      .= z - (y -' x) by A5,Def2;
  end;
end;
