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 m,REAL
 st f is_continuous_on Z & g is_continuous_on Z & Z c= dom f & Z c= dom g
 holds f(#)g is_continuous_on Z
proof
   let f,g be PartFunc of REAL m,REAL;
   assume A1: f is_continuous_on Z & g is_continuous_on Z;
   assume A2: Z c= dom f & Z c= dom g;
   reconsider f1=f, g1=g as PartFunc of REAL-NS m,REAL
      by REAL_NS1:def 4;
A3:Z c= dom f1 /\ dom g1 by A2,XBOOLE_1:19;
A4:dom (f1(#)g1) = dom f1 /\ dom g1 by VALUED_1:def 4;
   now let s1 be sequence of REAL-NS m;
    assume A5:rng s1 c= Z & s1 is convergent & lim s1 in Z; then
A6:f1/*s1 is convergent & g1/*s1 is convergent by A2,Th43,A1; then
A7:(f1/*s1)(#)(g1/*s1) is convergent;
A8:rng s1 c= dom f1 /\ dom g1 by A3,A5;
    hence (f1(#)g1)/*s1 is convergent by A7,RFUNCT_2:8;
    set y = lim s1;
    f1.y =f1/.y & g1.y =g1/.y by A5,A2,PARTFUN1:def 6; then
A9: f1.y = lim (f1/*s1) & g1.y = lim (g1/*s1) by A5,A2,Th43,A1;
    thus (f1(#)g1)/.(lim s1) = (f1(#)g1).y by A5,A3,A4,PARTFUN1:def 6
     .= f1.y * g1.y by VALUED_1:5
     .= lim ((f1/*s1)(#)(g1/*s1)) by A9,A6,SEQ_2:15
     .= lim ((f1(#)g1)/*s1) by A8,RFUNCT_2:8;
   end;
   hence thesis by A3,A4,Th43;
end;
