let n be Element of NAT ; :: thesis: for x0 being Real
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; :: 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 A1: ( 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,(REAL-NS n) by REAL_NS1:def 4;
reconsider g2 = f2 as PartFunc of (REAL-NS n),REAL by REAL_NS1:def 4;
f1 /. x0 = g1 /. x0 by REAL_NS1:def 4;
then g2 is_continuous_in g1 /. x0 by A1;
hence f2 * f1 is_continuous_in x0 by A1, Th20; :: thesis: verum