
theorem PT:
  for n be Nat holds
    Newton_Coeff (n + 1) = (<*0*>^(Newton_Coeff n))+((Newton_Coeff n)^<*0*>)
  proof
    let n be Nat;
    reconsider f = <*0*>^(Newton_Coeff n) as FinSequence of COMPLEX
      by NEWTON02:103;
    reconsider g = (Newton_Coeff n)^<*0*> as FinSequence of COMPLEX
      by NEWTON02:103;
    A1: len (Newton_Coeff (n + 1)) = n + 2 &
      n + 2 = len ((<*0*>^(Newton_Coeff n))+((Newton_Coeff n)^<*0*>))
        by CARD_1:def 7;
    for i be Nat st 1 <= i <= len (Newton_Coeff (n + 1)) holds
      (Newton_Coeff (n + 1)).i =
        ((<*0*>^(Newton_Coeff n))+((Newton_Coeff n)^<*0*>)).i
    proof
      let i be Nat such that
      B1: 1 <= i <= len (Newton_Coeff (n + 1));
      reconsider j = i - 1 as Nat by B1;
      B2: 1 <= i <= n + 1 + 1 by B1,CARD_1:def 7; then
      i in Seg (n + 1 + 1); then
      B3: i in dom Newton_Coeff (n+1) &
        i in dom ((<*0*>^(Newton_Coeff n))+((Newton_Coeff n)^<*0*>))
          by A1,FINSEQ_1:def 3;
      per cases by B2,XXREAL_0:1;
      suppose
        i = 1; then
        reconsider j as zero Nat;
        C2: (<*0*>^(Newton_Coeff n)).(j + 1) = 0;
        1 + 0 <= 1 + n by XREAL_1:6; then
        1 in Seg (len (Newton_Coeff (n+1-1))); then
        1 in dom Newton_Coeff n by FINSEQ_1:def 3; then
        ((Newton_Coeff n)^<*0*>).(j + 1) = (Newton_Coeff n).(j+1)
          by FINSEQ_1:def 7
        .= 1;
        hence thesis by B3,C2,VALUED_1:def 1;
      end;
      suppose
        C0: 1 < i < n + 1 + 1; then
        C1: 1 + 1 <= j + 1 <= n + 1 + 1 by INT_1:7;
        reconsider j as non zero Nat by C0;
        1 <= j <= len (Newton_Coeff (n+1-1)) by C1,XREAL_1:6; then
        j in dom Newton_Coeff n by FINSEQ_3:25; then
        C3: (Newton_Coeff n).j = n choose (j-1) by NEWTON:def 5;
        1 <= j + 1 <= len (Newton_Coeff (n+1-1)) by C0,INT_1:7; then
        C4: j + 1 in dom Newton_Coeff n by FINSEQ_3:25;
        (<*0*>^(Newton_Coeff n)).(j + 1) = (Newton_Coeff n).j &
        ((Newton_Coeff n)^<*0*>).(j + 1) = (Newton_Coeff n).(j + 1)
          by CF,FINSEQ1D7; then
        f.(j + 1) + g.(j + 1) =
          (n choose (j - 1)) + (n choose ((j+1)-1)) by C3,NEWTON:def 5,C4
        .= (n + 1) choose ((j-1)+1) by NEWTON:22
        .= (Newton_Coeff (n+1)).(j + 1) by B3,NEWTON:def 5;
        hence thesis by VALUED_1:def 1,B3;
      end;
      suppose
        C1: i = n + 2; then
        j = n + 1; then
        reconsider j as non zero Nat;
        C3: (<*0*>^(Newton_Coeff n)).(1 + j)
        = (Newton_Coeff n).(n+1) by C1,CF
        .= 1;
        (f+g).(j + 1) = f.(j + 1) + g.(j + 1) by B3,VALUED_1:def 1
        .= (Newton_Coeff (n+1)).((n+1)+1) by C1,C3;
        hence thesis by C1;
      end;
    end;
    hence thesis by A1;
  end;
