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

theorem
for f be PartFunc of REAL m,REAL, r be Real
  st Z c= dom f & f is_continuous_on Z holds r(#)f is_continuous_on Z
proof
   let f be PartFunc of REAL m,REAL, r be Real;
   assume Z c= dom f & f is_continuous_on Z; then
   <>*f is_continuous_on Z by Th44; then
A1:r(#)(<>*f) is_continuous_on Z by Th34;
   r(#)(<>*f) = <>*(r(#)f) by Th8;
   hence thesis by A1,Th44;
end;
