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

theorem Th13:
  z <=' y implies x + (y -' z) = x + y -' z
proof
  assume
A1: z <=' y;
  y <=' x + y by ARYTM_2:19;
  then
A2: z <=' x + y by A1,Th3;
  x + (y -' z) + z = x + ((y -' z) + z) by ARYTM_2:6
    .= x + y by A1,Def1
    .= x + y -' z + z by A2,Def1;
  hence thesis by ARYTM_2:11;
end;
