let X, Y be set ; :: thesis:
let C be non empty set ; :: thesis:
let f1, f2 be PartFunc of C,REAL; :: thesis:
thus ( f1 | X is bounded_above & f2 | Y is constant implies (f1 - f2) | (X /\ Y) is bounded_above ) :: thesis:

assume that
A1: f1 | X is bounded_above and
A2: f2 | Y is constant ; :: thesis:
(- f2) | Y is constant by A2, Th90;
hence (f1 - f2) | (X /\ Y) is bounded_above by A1, Th83; :: thesis:

end;

thus ( f1 | X is bounded_below & f2 | Y is constant implies (f1 - f2) | (X /\ Y) is bounded_below ) :: thesis:

assume that
A3: f1 | X is bounded_below and
A4: f2 | Y is constant ; :: thesis:
(- f2) | Y is constant by A4, Th90;
hence (f1 - f2) | (X /\ Y) is bounded_below by A3, Th83; :: thesis:

end;

assume that
A5: f1 | X is bounded and
A6: f2 | Y is constant ; :: thesis:
(- f2) | Y is constant by A6, Th90;
hence (f1 - f2) | (X /\ Y) is bounded by A5, Th83; :: thesis:
thus (f2 - f1) | (X /\ Y) is bounded by A5, A6, Th84; :: thesis:
thus (f1 (#) f2) | (X /\ Y) is bounded by A5, A6, Th84; :: thesis: