reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th32:
for f be PartFunc of REAL m,REAL n, x be Element of REAL m
  st f is_continuous_in x holds |.f.| is_continuous_in x
proof
   let f be PartFunc of REAL m,REAL n, x be Element of REAL m;
   assume A1: f is_continuous_in x;
   reconsider y=x as Point of REAL-NS m by REAL_NS1:def 4;
A2:the carrier of REAL-NS m = REAL m &
   the carrier of REAL-NS n = REAL n by REAL_NS1:def 4; then
   reconsider f1=f as PartFunc of REAL-NS m,REAL-NS n;
A3: ||.f1.|| is_continuous_in y by NFCONT_1:17,A1;
   |.f.|=||.f1.|| by A2,NFCONT_4:9;
   hence thesis by A3,NFCONT_4:21;
end;
