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

theorem Th8:
for f be PartFunc of REAL i,REAL, r be Real
 holds <>*(r(#)f) = r(#)(<>*f)
proof
   let f be PartFunc of REAL i,REAL, r be Real;
A1:dom (<>*(r(#)f)) = dom (r(#)f) by Th3; then
A2:dom (<>*(r(#)f)) = dom f by VALUED_1:def 5; then
A3:dom (<>*(r(#)f)) = dom (<>*f) by Th3
                   .= dom (r(#)(<>*f)) by VALUED_2:def 39;
   now let x be object;
    reconsider fx = f.x as Element of REAL by XREAL_0:def 1;
    assume A4:x in dom (<>*(r(#)f)); then
    (<>*(r(#)f)).x = <* (r(#)f).x *> by A1,Th6
                  .= <* r*(f.x) *> by A4,A1,VALUED_1:def 5
                  .= r(#)<* fx *> by RVSUM_1:47
                  .= r(#)(<>*f).x by A4,A2,Th6;
    hence (<>*(r(#)f)).x = (r(#)(<>*f)).x by A4,A3,VALUED_2:def 39;
   end;
   hence thesis by A3,FUNCT_1:2;
end;
