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

proof

assume A1: ( f1 | X is bounded_above & f2 | Y is constant ) ; :: thesis: (f1 - f2) | (X /\ Y) is bounded_above
then (- f2) | Y is constant by Th107;
hence (f1 - f2) | (X /\ Y) is bounded_above by A1, Th100; :: thesis: verum

end;

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

proof

assume A2: ( f1 | X is bounded_below & f2 | Y is constant ) ; :: thesis: (f1 - f2) | (X /\ Y) is bounded_below
then (- f2) | Y is constant by Th107;
hence (f1 - f2) | (X /\ Y) is bounded_below by A2, Th100; :: thesis: verum

end;

assume A3: ( f1 | X is bounded & f2 | Y is constant ) ; :: thesis: ( (f1 - f2) | (X /\ Y) is bounded & (f2 - f1) | (X /\ Y) is bounded & (f1 (#) f2) | (X /\ Y) is bounded )
then (- f2) | Y is constant by Th107;
hence (f1 - f2) | (X /\ Y) is bounded by A3, Th100; :: thesis: ( (f2 - f1) | (X /\ Y) is bounded & (f1 (#) f2) | (X /\ Y) is bounded )
thus (f2 - f1) | (X /\ Y) is bounded by A3, Th101; :: thesis: (f1 (#) f2) | (X /\ Y) is bounded
thus (f1 (#) f2) | (X /\ Y) is bounded by A3, Th101; :: thesis: verum