reserve f for Function;
reserve n,k,n1 for Nat;
reserve r,p for Real;
reserve x,y,z for object;
reserve seq,seq1,seq2,seq3,seq9,seq19 for Real_Sequence;

theorem Th19:
  r (#) (seq1 (#) seq2) = seq1 (#) (r (#) seq2)
proof
  now
    let n be Element of NAT;
    thus (r(#)(seq1(#)seq2)).n=r*(seq1(#)seq2).n by Th9
      .=r*(seq1.n*seq2.n) by Th8
      .=seq1.n*(r*seq2.n)
      .=seq1.n*(r(#)seq2).n by Th9
      .=(seq1(#)(r(#)seq2)).n by Th8;
  end;
  hence thesis by FUNCT_2:63;
end;
