
theorem NLM4:
  for c,c1 be non empty positive-yielding XFinSequence of REAL,
  a be Real
  st c1=a(#)c holds
  for x be Nat
  holds (polynom(c1)).x = a* ( (polynom(c)).x)
  proof
    let c,c1 be non empty positive-yielding XFinSequence of REAL,
    a be Real;
    assume AS: c1=a(#)c;
    let x be Nat;
    D1: dom (c1 (#) seq_a^(x,1,0)) = dom c1 by ASYMPT_2:26
    .= dom c by VALUED_1:def 5,AS;
    D2: dom (a (#) ( c (#) seq_a^(x,1,0) ))
     =dom ( c (#) seq_a^(x,1,0) ) by VALUED_1:def 5
    .= dom c by ASYMPT_2:26;
    for i be object
    st i in dom (c1 (#) seq_a^(x,1,0))
    holds
    (c1 (#) seq_a^(x,1,0)).i = (a (#) ( c (#) seq_a^(x,1,0) )).i
    proof
      let i be object;
      assume D3: i in dom (c1 (#) seq_a^(x,1,0));
      then
      D4: i in dom c1 by ASYMPT_2:26;
      thus (c1 (#) seq_a^(x,1,0)).i
      = c1.i * (seq_a^(x,1,0)).i by D4,ASYMPT_2:26
      .= a*c.i * (seq_a^(x,1,0)).i by AS,VALUED_1:6
      .= a* ( c.i * (seq_a^(x,1,0)).i )
      .= a* ( c (#) seq_a^(x,1,0) ).i by D3,D1,ASYMPT_2:26
      .= (a (#) ( c (#) seq_a^(x,1,0) )).i by VALUED_1:6;
    end;
    then
    P2: c1 (#) seq_a^(x,1,0) = a (#) ( c (#) seq_a^(x,1,0) ) by D1,D2;
    thus (polynom(c1)).x = Sum( c1 (#) seq_a^(x,1,0)) by ASYMPT_2:def 2
    .= a* Sum( c (#) seq_a^(x,1,0)) by AFINSQ_2:64,P2
    .= a* (polynom(c)).x by ASYMPT_2:def 2;
  end;
