reserve k, m, n, p, K, N for Nat;
reserve i for Integer;
reserve x, y, eps for Real;
reserve seq, seq1, seq2 for Real_Sequence;
reserve sq for FinSequence of REAL;

theorem Th19:
  k<=n implies ((x,y) In_Power n).(k+1)=(n choose k)*(x ^ (n-k))*( y ^ k)
proof
  reconsider i1 = (k+1)-1 as Element of NAT by ORDINAL1:def 12;
A1: k+1>=0+1 & len((x,y) In_Power n)=n+1 by NEWTON:def 4,XREAL_1:6;
  assume
A2: k<=n;
  then reconsider l = n-i1 as Element of NAT by INT_1:5;
  k+1<=n+1 by A2,XREAL_1:6;
  then k+1 in dom((x,y) In_Power n) by A1,FINSEQ_3:25;
  hence ((x,y) In_Power n).(k+1) = (n choose i1)*(x ^ l)*(y|^i1) by
NEWTON:def 4
    .= (n choose k)*(x ^ (n-k))*(y ^ k);
end;
