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 Th4:
for h be PartFunc of REAL,the carrier of S,
    seq be Real_Sequence, r be Real
 st rng seq c= dom h holds (r(#)h)/*seq = r*(h/*seq)
proof
   let h be PartFunc of REAL,the carrier of S,
       seq be Real_Sequence, 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;
A3: n in NAT by ORDINAL1:def 12;
A4: seq.n in dom (r(#)h) by A2,Th1;
    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;
