
theorem LmN:
  for n,k be Nat st k in Seg (n+1) holds ex l,m be Nat st
  l = k - 1 & m = n - l
  proof
    let n,k be Nat such that
    A1: k in Seg (n+1);
    reconsider k as non zero Nat by A1,FINSEQ_1:1;
    reconsider l = k - 1 as Nat;
    (n - (k - 1)) + k >= 0 + k by A1,FINSEQ_1:1; then
    n - l >= 0 by XREAL_1:6;
    hence thesis;
  end;
