let n be Element of NAT ; :: thesis: for x0 being real number
for f1 being PartFunc of REAL,(REAL n)
for f2 being PartFunc of (REAL n),REAL st x0 in dom (f2 * f1) & f1 is_continuous_in x0 & f2 is_continuous_in f1 /. x0 holds
f2 * f1 is_continuous_in x0
let x0 be real number ; :: thesis: for f1 being PartFunc of REAL,(REAL n)
for f2 being PartFunc of (REAL n),REAL st x0 in dom (f2 * f1) & f1 is_continuous_in x0 & f2 is_continuous_in f1 /. x0 holds
f2 * f1 is_continuous_in x0
let f1 be PartFunc of REAL,(REAL n); :: thesis: for f2 being PartFunc of (REAL n),REAL st x0 in dom (f2 * f1) & f1 is_continuous_in x0 & f2 is_continuous_in f1 /. x0 holds
f2 * f1 is_continuous_in x0
let f2 be PartFunc of (REAL n),REAL; :: thesis: ( x0 in dom (f2 * f1) & f1 is_continuous_in x0 & f2 is_continuous_in f1 /. x0 implies f2 * f1 is_continuous_in x0 )
assume AS: ( x0 in dom (f2 * f1) & f1 is_continuous_in x0 & f2 is_continuous_in f1 /. x0 ) ; :: 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;
reconsider g2 = f2 as PartFunc of the carrier of (REAL-NS n),REAL by REAL_NS1:def 4;
A1: g1 is_continuous_in x0 by AS, Def1Th;
f1 /. x0 = g1 /. x0 by REAL_NS1:def 4;
then g2 is_continuous_in g1 /. x0 by AS, Def1aTh;
hence f2 * f1 is_continuous_in x0 by A1, AS, Th12Y; :: thesis: verum