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

theorem
  y <=' x & y <=' z implies x + (z -' y) = x -' y + z
proof
  assume that
A1: y <=' x and
A2: y <=' z;
  x + (z -' y) + y = x + ((z -' y) + y) by ARYTM_2:6
    .= x + z by A2,Def1
    .= x -' y + y + z by A1,Def1
    .= x -' y + z + y by ARYTM_2:6;
  hence thesis by ARYTM_2:11;
end;
