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 ) ) )

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 A6, Th90;

hence (f1 - f2) | (X /\ Y) is bounded by A5, Th83; :: 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

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

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 ) )
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 A2, Th90;

hence (f1 - f2) | (X /\ Y) is bounded_above by A1, Th83; :: thesis: verum

end;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 A2, Th90;

hence (f1 - f2) | (X /\ Y) is bounded_above by A1, Th83; :: thesis: verum

proof

assume that
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 A4, Th90;

hence (f1 - f2) | (X /\ Y) is bounded_below by A3, Th83; :: thesis: verum

end;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 A4, Th90;

hence (f1 - f2) | (X /\ Y) is bounded_below by A3, Th83; :: thesis: verum

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 A6, Th90;

hence (f1 - f2) | (X /\ Y) is bounded by A5, Th83; :: 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