reserve n,m,k for Nat;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,g,x0,x1,x2 for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of REAL,the carrier of S;
reserve s1,s2 for Real_Sequence;
reserve Y for Subset of REAL;

theorem
for h be sequence of S,r be Real holds r(#)h = r*h
proof
   let h be sequence of S;
   let r be Real;
A1: dom h = NAT by FUNCT_2:def 1; then
A2:dom (r(#)h) = NAT by VFUNCT_1:def 4;
   now let n;
A3: n in NAT by ORDINAL1:def 12;
    then (r(#)h).n = (r(#)h)/.n by A2,PARTFUN1:def 6; then
    (r(#)h).n = r * (h/.n) by A2,VFUNCT_1:def 4,A3;
    hence (r(#)h).n = r * (h.n) by A3,PARTFUN1:def 6,A1;
   end;
   hence thesis by NORMSP_1:def 5;
end;
