reserve x,X,Y for set;
reserve g,r,r1,r2,p,p1,p2 for Real;
reserve R for Subset of REAL;
reserve seq,seq1,seq2,seq3 for Real_Sequence;
reserve Ns for increasing sequence of NAT;
reserve n for Nat;
reserve W for non empty set;
reserve h,h1,h2 for PartFunc of W,REAL;

theorem Th5:
  (seq*Ns)" = (seq")*Ns
proof
  now
    let n be Element of NAT;
    thus ((seq*Ns)").n = ((seq*Ns).n)" by VALUED_1:10
      .= (seq.(Ns.n))" by FUNCT_2:15
      .= (seq").(Ns.n) by VALUED_1:10
      .= ((seq")*Ns).n by FUNCT_2:15;
  end;
  hence thesis by FUNCT_2:63;
end;
