let x, y, t be ExtReal; :: thesis: ( t <> -infty & t <> +infty & x < y implies x + t < y + t )
assume that
A1: ( t <> -infty & t <> +infty ) and
A2: x < y ; :: thesis: x + t < y + t
A3: t - t = 0 by Th7;

A4: now :: thesis: not x + t = y + t

assume x + t = y + t ; :: thesis: contradiction
then x + (t - t) = (y + t) - t by A1, Th30;
then x + 0 = y + (t - t) by A1, A3, Th30;
then x = y + 0 by A3, Th4;
hence contradiction by A2, Th4; :: thesis: verum

end;

x + t <= y + t by A2, Th36;
hence x + t < y + t by A4, XXREAL_0:1; :: thesis: verum