theorem
  for k, i being Integer holds
  k < i implies ex j being Element of NAT st j = i-k & 1 <= j
proof
  let k,i be Integer;
  assume k < i;
  then
A1: k - k < i - k by XREAL_1:9;
  then reconsider j = i - k as Element of NAT by Th3;
  reconsider j9 = j, Z9 = 0 as Integer;
  take j;
  thus j = i - k;
  Z9 + 1 <= j9 by A1,Th7;
  hence thesis;
end;
