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

theorem Th13:
  for h be PartFunc of S,T for seq be sequence of S
  for r be Real
  st rng seq c= dom h holds (r(#)h)/*seq = r*(h/*seq)
proof
  let h be PartFunc of S,T;
  let seq be sequence of S;
  let r be Real;
  assume
A1: rng seq c= dom h;
  then
A2: rng seq c= dom (r(#)h) by VFUNCT_1:def 4;
  now
    let n be Nat;
A3:   n in NAT by ORDINAL1:def 12;
A4: seq.n in dom (r(#)h) by A2,Th5;
    thus ((r(#)h)/*seq).n = (r(#)h)/.(seq.n) by A2,FUNCT_2:109,A3
      .= r * (h/.(seq.n)) by A4,VFUNCT_1:def 4
      .= r * (h/*seq).n by A1,FUNCT_2:109,A3;
  end;
  hence thesis by NORMSP_1:def 5;
end;
