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

theorem
  t <> -infty & t <> +infty & x < y implies x - t < y - t
proof
  assume that
A1: t <> -infty & t <> +infty and
A2: x < y;
A3: t-t = 0 by Th7;
A4: now
    assume x - t = y - t;
    then x - (t -t) = y - t+t by A1,Th32;
    then x - 0 = y - (t-t) by A1,A3,Th32;
    then x= y+ 0 by A3,Th4;
    hence contradiction by A2,Th4;
  end;
  x - t <= y - t by A2,Th37;
  hence thesis by A4,XXREAL_0:1;
end;
