reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve c for Complex;

theorem Th41:
  1 <= i <= len ((a,b) In_Power n) - m implies
  a|^m divides ((a,b) In_Power n).i
  proof
    set P = (a,b) In_Power n;
    assume that
A1: 1 <= i and
A2: i <= len P - m;
    reconsider M = i-1 as Element of NAT by A1,INT_1:5;
    len P - m <= len P by XREAL_1:43;
    then
A3: i <= len P by A2,XXREAL_0:2;
    then
A4: i in dom P by A1,FINSEQ_3:25;
A5: len P = n+1 by NEWTON:def 4;
    then i-1 <= n+1-1 by A3,XREAL_1:9;
    then reconsider L = n-M as Element of NAT by INT_1:5;
    P.i = (n choose M) * a|^L * b|^M by A4,NEWTON:def 4
    .= a|^L * ((n choose M)*b|^M);
    then
A6: a|^L divides P.i;
    m <= n+1-i by A2,A5,XREAL_1:11;
    then a|^m divides a|^L by NEWTON:89;
    hence a|^m divides P.i by A6,INT_2:9;
  end;
