let n be Element of NAT ; :: thesis: for x0 being real number
for S being RealNormSpace
for z being Point of (REAL-NS n)
for f1 being PartFunc of REAL,(REAL n)
for f2 being PartFunc of the carrier of (REAL-NS n), the carrier of S st x0 in dom (f2 * f1) & f1 is_continuous_in x0 & z = f1 /. x0 & f2 is_continuous_in z holds
f2 * f1 is_continuous_in x0
let x0 be real number ; :: thesis: for S being RealNormSpace
for z being Point of (REAL-NS n)
for f1 being PartFunc of REAL,(REAL n)
for f2 being PartFunc of the carrier of (REAL-NS n), the carrier of S st x0 in dom (f2 * f1) & f1 is_continuous_in x0 & z = f1 /. x0 & f2 is_continuous_in z holds
f2 * f1 is_continuous_in x0
let S be RealNormSpace; :: thesis: for z being Point of (REAL-NS n)
for f1 being PartFunc of REAL,(REAL n)
for f2 being PartFunc of the carrier of (REAL-NS n), the carrier of S st x0 in dom (f2 * f1) & f1 is_continuous_in x0 & z = f1 /. x0 & f2 is_continuous_in z holds
f2 * f1 is_continuous_in x0
let z be Point of (REAL-NS n); :: thesis: for f1 being PartFunc of REAL,(REAL n)
for f2 being PartFunc of the carrier of (REAL-NS n), the carrier of S st x0 in dom (f2 * f1) & f1 is_continuous_in x0 & z = f1 /. x0 & f2 is_continuous_in z holds
f2 * f1 is_continuous_in x0
let f1 be PartFunc of REAL,(REAL n); :: thesis: for f2 being PartFunc of the carrier of (REAL-NS n), the carrier of S st x0 in dom (f2 * f1) & f1 is_continuous_in x0 & z = f1 /. x0 & f2 is_continuous_in z holds
f2 * f1 is_continuous_in x0
let f2 be PartFunc of the carrier of (REAL-NS n), the carrier of S; :: thesis: ( x0 in dom (f2 * f1) & f1 is_continuous_in x0 & z = f1 /. x0 & f2 is_continuous_in z implies f2 * f1 is_continuous_in x0 )
assume AS: ( x0 in dom (f2 * f1) & f1 is_continuous_in x0 & z = f1 /. x0 & f2 is_continuous_in z ) ; :: thesis: f2 * f1 is_continuous_in x0
reconsider g1 = f1 as PartFunc of REAL, the carrier of (REAL-NS n) by REAL_NS1:def 4;
P1: g1 is_continuous_in x0 by AS, Def1Th;
f1 /. x0 = g1 /. x0 by REAL_NS1:def 4;
hence f2 * f1 is_continuous_in x0 by P1, AS, NFCONT_3:15; :: thesis: verum