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 Th2:
  (seq1 + seq2)*Ns = (seq1*Ns) + (seq2*Ns) & (seq1 - seq2)*Ns = (
  seq1*Ns) - (seq2*Ns) & (seq1 (#) seq2)*Ns = (seq1*Ns) (#) (seq2*Ns)
proof
  now
    let n be Element of NAT;
    thus ((seq1 + seq2)*Ns).n = (seq1 + seq2).(Ns.n) by FUNCT_2:15
      .= seq1.(Ns.n) + seq2.(Ns.n) by SEQ_1:7
      .= (seq1*Ns).n + seq2.(Ns.n) by FUNCT_2:15
      .= (seq1*Ns).n + (seq2*Ns).n by FUNCT_2:15
      .= (seq1*Ns + seq2*Ns).n by SEQ_1:7;
  end;
  hence (seq1 + seq2)*Ns = (seq1*Ns) + (seq2*Ns) by FUNCT_2:63;
  now
    let n be Element of NAT;
    thus ((seq1 - seq2)*Ns).n = (seq1 - seq2).(Ns.n) by FUNCT_2:15
      .= seq1.(Ns.n) - seq2.(Ns.n) by Th1
      .= (seq1*Ns).n - seq2.(Ns.n) by FUNCT_2:15
      .= (seq1*Ns).n - (seq2*Ns).n by FUNCT_2:15
      .= (seq1*Ns - seq2*Ns).n by Th1;
  end;
  hence (seq1 - seq2)*Ns = (seq1*Ns) - (seq2*Ns) by FUNCT_2:63;
  now
    let n be Element of NAT;
    thus ((seq1 (#) seq2)*Ns).n = (seq1 (#) seq2).(Ns.n) by FUNCT_2:15
      .= seq1.(Ns.n) * seq2.(Ns.n) by SEQ_1:8
      .= (seq1*Ns).n * seq2.(Ns.n) by FUNCT_2:15
      .= (seq1*Ns).n * (seq2*Ns).n by FUNCT_2:15
      .= ((seq1*Ns)(#)(seq2*Ns)).n by SEQ_1:8;
  end;
  hence thesis by FUNCT_2:63;
end;
