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 that
A1: f1 | X is bounded_above and
A2: f2 | Y is constant ; :: thesis: (f1 - f2) | (X /\ Y) is bounded_above
(- f2) | Y is constant by ;
hence (f1 - f2) | (X /\ Y) is bounded_above by ; :: 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 that
A3: f1 | X is bounded_below and
A4: f2 | Y is constant ; :: thesis: (f1 - f2) | (X /\ Y) is bounded_below
(- f2) | Y is constant by ;
hence (f1 - f2) | (X /\ Y) is bounded_below by ; :: thesis: verum
end;
assume that
A5: f1 | X is bounded and
A6: f2 | Y is constant ; :: thesis: ( (f1 - f2) | (X /\ Y) is bounded & (f2 - f1) | (X /\ Y) is bounded & (f1 (#) f2) | (X /\ Y) is bounded )
(- f2) | Y is constant by ;
hence (f1 - f2) | (X /\ Y) is bounded by ; :: thesis: ( (f2 - f1) | (X /\ Y) is bounded & (f1 (#) f2) | (X /\ Y) is bounded )
thus (f2 - f1) | (X /\ Y) is bounded by A5, A6, Th84; :: thesis: (f1 (#) f2) | (X /\ Y) is bounded
thus (f1 (#) f2) | (X /\ Y) is bounded by A5, A6, Th84; :: thesis: verum