reserve x,y,z,w for ExtReal,
  r for Real;
reserve f,g for ExtReal;

theorem Th43:
  z in REAL & x < y implies x + z < y + z & x - z < y - z
proof
  assume that
A1: z in REAL and
A2: x < y;
A3: x + z <> y + z
  proof
    assume x + z = y + z;
    then x = (y + z) - z by A1,Th22
      .= y by A1,Th22;
    hence thesis by A2;
  end;
A4: x - z <> y - z
  proof
    assume x - z = y - z;
    then x = (y - z) + z by A1,Th22
      .= y by A1,Th22;
    hence thesis by A2;
  end;
  x + z <= y + z & x - z <= y - z by A2,Th36;
  hence thesis by A3,A4,XXREAL_0:1;
end;
