
theorem LMStat2:
  for k,i,j,i0,j0 be Nat st 1 <= k & k <= 128 & i in Seg 4 & j in Seg 4 &
  i0 in Seg 4 & j0 in Seg 4 & 1+(i-'1)*8+(j-'1)*32 <= k &
  k <= 1+(i-'1)*8+(j-'1)*32+7 & 1+(i0-'1)*8+(j0-'1)*32 <= k &
  k <= 1+(i0-'1)*8+(j0-'1)*32+7 holds i = i0 & j = j0
proof
  let k,i,j,i0,j0 be Nat;
  assume
AS: 1 <= k & k <= 128 & i in Seg 4 & j in Seg 4 & i0 in Seg 4 & j0 in Seg 4 &
  1+(i-'1)*8+(j-'1)*32 <= k & k <= 1+(i-'1)*8+(j-'1)*32+7 &
  1+(i0-'1)*8+(j0-'1)*32 <= k & k <= 1+(i0-'1)*8+(j0-'1)*32+7;
  assume not (i = i0 & j = j0);
  then
A2: {n where n is Nat : 1+(i-'1)*8+(j-'1)*32 <= n & n <= 8+(i-'1)*8+(j-'1)*32}
  /\ {n where n is Nat : 1+(i0-'1)*8+(j0-'1)*32 <= n &
  n <= 8+(i0-'1)*8+(j0-'1)*32} = {} by LMStat2A,AS;
A3: k in {n where n is Nat : 1+(i-'1)*8+(j-'1)*32 <= n &
  n <= 8+(i-'1)*8+(j-'1)*32} by AS;
  k in {n where n is Nat : 1+(i0-'1)*8+(j0-'1)*32 <= n &
  n <= 8+(i0-'1)*8+(j0-'1)*32} by AS;
  hence contradiction by A3,XBOOLE_0:def 4,A2;
end;
