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

theorem Th30:
for f be PartFunc of REAL m,REAL n, x be Element of REAL m, r be Real
 st f is_continuous_in x holds r(#)f is_continuous_in x
proof
   let f be PartFunc of REAL m,REAL n, x be Element of REAL m, r be Real;
   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:r(#)f1 is_continuous_in y by NFCONT_1:16,A1;
   r(#)f=r(#)f1 by A2,NFCONT_4:6;
   hence thesis by A3;
end;
