reserve n for non zero Nat,
  j,k,l,m for Nat,
  g,h,i for Integer;

theorem Th8:
  l + m <= k - 1 implies l < k & m < k
proof
  assume
A1: l + m <= k - 1;
  then
A2: l + m - m <= k - 1 - m by XREAL_1:9;
  k <= k + m by NAT_1:11;
  then
A3: k - m <= k + m - m by XREAL_1:9;
  k - 1 - m = k - m - 1;
  then k - 1 - m < k by A3,XREAL_1:146,XXREAL_0:2;
  hence l < k by A2,XXREAL_0:2;
  k <= k + l by NAT_1:11;
  then
A4: k - l <= k + l - l by XREAL_1:9;
A5: m + l - l <= k - 1 - l by A1,XREAL_1:9;
  k - 1 - l = k - l - 1;
  then k - 1 - l < k by A4,XREAL_1:146,XXREAL_0:2;
  hence thesis by A5,XXREAL_0:2;
end;
