let X, Y be set ; :: thesis: for C being non empty set
for f1, f2 being PartFunc of C,COMPLEX st f1 | X is bounded & f2 | Y is constant holds
( (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,COMPLEX st f1 | X is bounded & f2 | Y is constant holds
( (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,COMPLEX; :: 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 and
A2: 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 A2, Th79;
hence (f1 - f2) | (X /\ Y) is bounded by A1, Th82; :: thesis: ( (f2 - f1) | (X /\ Y) is bounded & (f1 (#) f2) | (X /\ Y) is bounded )
A3: f2 | Y is bounded by A2, Th80;
hence (f2 - f1) | (X /\ Y) is bounded by A1, Th75; :: thesis: (f1 (#) f2) | (X /\ Y) is bounded
thus (f1 (#) f2) | (X /\ Y) is bounded by A1, A3, Th75; :: thesis: verum