let X, Y be set ; 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 ; 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; ( 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
; ( (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; ( (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; (f1 (#) f2) | (X /\ Y) is bounded
thus
(f1 (#) f2) | (X /\ Y) is bounded
by A1, A3, Th75; verum