let x, y, t be ext-real number ; :: thesis: ( t <> -infty & t <> +infty & x < y implies x - t < y - t )
assume A1: ( t <> -infty & t <> +infty & x < y ) ; :: thesis: x - t < y - t
then A2: x - t <= y - t by Th37;
A3: t - t = 0 by Tx4;

now

assume x - t = y - t ; :: thesis: contradiction
then x - (t - t) = (y - t) + t by A1, Th109;
then x - 0 = y - (t - t) by A1, A3, Th109;
then x = y + 0 by A3, Tx3;
hence contradiction by A1, Tx3; :: thesis: verum

end;

hence x - t < y - t by A2, XXREAL_0:1; :: thesis: verum