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 Th3:
  (p(#)seq)*Ns = p(#)(seq*Ns)
proof
  now
    let n be Element of NAT;
    thus ((p(#)seq)*Ns).n = (p(#)seq).(Ns.n) by FUNCT_2:15
      .= p*(seq.(Ns.n)) by SEQ_1:9
      .= p*((seq*Ns).n) by FUNCT_2:15
      .= (p(#)(seq*Ns)).n by SEQ_1:9;
  end;
  hence thesis by FUNCT_2:63;
end;
