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

theorem
for f,g be PartFunc of REAL i,REAL, x be Element of REAL i
 st f is_continuous_in x & g is_continuous_in x holds f(#)g is_continuous_in x
proof
   let g1,g2 be PartFunc of REAL i,REAL, x be Element of REAL i;
   assume A1: g1 is_continuous_in x & g2 is_continuous_in x;
   reconsider y=x as Point of REAL-NS i by REAL_NS1:def 4;
   reconsider f1=g1, f2=g2 as PartFunc of REAL-NS i,REAL
      by REAL_NS1:def 4;
A2:dom (f1(#)f2) = dom f1 /\ dom f2 by VALUED_1:def 4;
   f1 is_continuous_in y & f2 is_continuous_in y by A1,NFCONT_4:21; then
A3:y in dom f1 & y in dom f2 by NFCONT_1:def 6; then
A4:y in dom (f1(#)f2) by A2,XBOOLE_0:def 4;
   now let s1 be sequence of REAL-NS i;
    assume that
A5: rng s1 c= dom(f1(#)f2) and
A6: s1 is convergent & lim s1=y;
    dom (f1(#)f2) c= dom f1 & dom (f1(#)f2) c= dom f2 by A2,XBOOLE_1:17; then
A7:rng s1 c= dom f1 & rng s1 c= dom f2 by A5; then
A8:f1/*s1 is convergent & f2/*s1 is convergent by A1,A6,Lm3; then
    (f1/*s1)(#)(f2/*s1) is convergent;
    hence (f1(#)f2)/*s1 is convergent by A2,A5,RFUNCT_2:8;
    f1.y = f1/.y & f2.y = f2/.y by A3,PARTFUN1:def 6; then
A9:f1.y = lim (f1/*s1) & f2.y = lim (f2/*s1) by A1,A6,A7,Lm3;
    thus (f1(#)f2)/.y = (f1(#)f2).y by A4,PARTFUN1:def 6
      .= f1.y * f2.y by VALUED_1:5
      .= lim ((f1/*s1)(#)(f2/*s1)) by A9,A8,SEQ_2:15
      .= lim ((f1(#)f2)/*s1) by A2,A5,RFUNCT_2:8;
   end;
   hence thesis by Lm3,A4;
end;
