let x, y, z be ext-real number ; :: thesis: ( ( x = +infty & y = +infty ) or ( x = -infty & y = -infty ) or not x - y < z or ( x <> +infty & y <> -infty & z <> -infty & x < z + y ) )
assume that
A1: ( not ( x = +infty & y = +infty ) & not ( x = -infty & y = -infty ) ) and
A2: x - y < z ; :: thesis: ( x <> +infty & y <> -infty & z <> -infty & x < z + y )
per cases ( z = +infty or z <> +infty ) ;
suppose A3: z = +infty ; :: thesis: ( x <> +infty & y <> -infty & z <> -infty & x < z + y )
then ( x <> +infty & y <> -infty & z <> -infty ) by A1, A2, Th16, Th29;
then ( x < +infty & z + y = +infty ) by A3, XXREAL_0:4, Def3;
hence ( x <> +infty & y <> -infty & z <> -infty & x < z + y ) by A3; :: thesis: verum
end;
suppose A4: z <> +infty ; :: thesis: ( x <> +infty & y <> -infty & z <> -infty & x < z + y )
x - y < +infty by A2, XXREAL_0:2, XXREAL_0:4;
then A5: ( x <> +infty & y <> -infty ) by A1, Th16, Th29;
A6: -infty <= x - y by XXREAL_0:5;
then z in REAL by A2, A4, XXREAL_0:14;
then reconsider c = z as real number ;
per cases ( y = +infty or y <> +infty ) ;
suppose A7: y = +infty ; :: thesis: ( x <> +infty & y <> -infty & z <> -infty & x < z + y )
then x < +infty by A1, XXREAL_0:4;
hence ( x <> +infty & y <> -infty & z <> -infty & x < z + y ) by A2, A6, A7, Def3; :: thesis: verum
end;
suppose y <> +infty ; :: thesis: ( x <> +infty & y <> -infty & z <> -infty & x < z + y )
then y in REAL by A5, XXREAL_0:14;
then reconsider b = y as real number ;
per cases ( x = -infty or x <> -infty ) ;
suppose x <> -infty ; :: thesis: ( x <> +infty & y <> -infty & z <> -infty & x < z + y )
then x in REAL by A5, XXREAL_0:14;
then reconsider a = x as real number ;
A9: ( x - y = a - b & z = c ) ;
a < c + b by A2, A9, XREAL_1:21;
hence ( x <> +infty & y <> -infty & z <> -infty & x < z + y ) ; :: thesis: verum
end;
end;
end;
end;
end;
end;