let n be Element of NAT ; :: thesis: for X, X1 being set
for f1, f2 being PartFunc of REAL,(REAL n) 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 ; :: thesis: for f1, f2 being PartFunc of REAL,(REAL n) 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,(REAL n); :: thesis: ( 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 AS: ( X c= dom f1 & X1 c= dom f2 & f1 | X is continuous & f2 | X1 is continuous ) ; :: thesis: ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous )
reconsider g1 = f1, g2 = f2 as PartFunc of REAL, the carrier of (REAL-NS n) by REAL_NS1:def 4;
P2: g1 | X is continuous by AS, Def2Th;
g2 | X1 is continuous by AS, Def2Th;
then P3: ( (g1 + g2) | (X /\ X1) is continuous & (g1 - g2) | (X /\ X1) is continuous ) by AS, P2, NFCONT_3:20;
P4: g1 + g2 = f1 + f2 by LM003A;
g1 - g2 = f1 - f2 by LM003D;
hence ( (f1 + f2) | (X /\ X1) is continuous & (f1 - f2) | (X /\ X1) is continuous ) by P3, P4, Def2Th; :: thesis: verum