let S be RealNormSpace; for X, X1 being set
for f1, f2 being PartFunc of REAL, the carrier of S st X c= dom f1 & X1 c= dom f2 & f1 | X is continuous & f2 | X1 is continuous holds
( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous )
let X, X1 be set ; for f1, f2 being PartFunc of REAL, the carrier of S st X c= dom f1 & X1 c= dom f2 & f1 | X is continuous & f2 | X1 is continuous holds
( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous )
let f1, f2 be PartFunc of REAL, the carrier of S; ( X c= dom f1 & X1 c= dom f2 & f1 | X is continuous & f2 | X1 is continuous implies ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous ) )
assume
( X c= dom f1 & X1 c= dom f2 )
; ( not f1 | X is continuous or not f2 | X1 is continuous or ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous ) )
then A1:
X /\ X1 c= (dom f1) /\ (dom f2)
by XBOOLE_1:27;
assume
( f1 | X is continuous & f2 | X1 is continuous )
; ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous )
then
( f1 | (X /\ X1) is continuous & f2 | (X /\ X1) is continuous )
by Th18, XBOOLE_1:17;
hence
( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous )
by A1, Th19; verum