
theorem LMStat2A:
  for i,j,i0,j0 be Nat st i in Seg 4 & j in Seg 4 & i0 in Seg 4 &
  j0 in Seg 4 & not (i = i0 & j = j0)
  holds {k where k is Nat : 1+(i-'1)*8+(j-'1)*32 <= k &
  k <= 8+(i-'1)*8+(j-'1)*32} /\ {k where k is Nat :
  1+(i0-'1)*8+(j0-'1)*32 <= k & k <= 8+(i0-'1)*8+(j0-'1)*32} = {}
proof
  let i,j,i0,j0 be Nat;
  assume
AS: i in Seg 4 & j in Seg 4 & i0 in Seg 4 &
  j0 in Seg 4 & not (i = i0 & j = j0);
  set A = {k where k is Nat : 1+(i-'1)*8+(j-'1)*32 <= k &
  k <= 8+(i-'1)*8+(j-'1)*32};
  set B = {k where k is Nat : 1+(i0-'1)*8+(j0-'1)*32 <= k &
  k <= 8+(i0-'1)*8+(j0-'1)*32};
A1: 1 <= j & j <= 4 by AS,FINSEQ_1:1;
A2: 1 <= i & i <= 4 by AS,FINSEQ_1:1;
B1: 1 <= j0 & j0 <= 4 by AS,FINSEQ_1:1;
B2: 1 <= i0 & i0 <= 4 by AS,FINSEQ_1:1;
P1: (j-'1) = j-1 by XREAL_1:233,A1;
P2: (i-'1) = i-1 by XREAL_1:233,A2;
P3: (j0-'1) = j0-1 by XREAL_1:233,B1;
P4: (i0-'1) = i0-1 by XREAL_1:233,B2;
  i-1 <= 4-1 by A2,XREAL_1:9;
  then
R2: i-'1 <= 3 by XREAL_1:233,A2;
  i0-1 <= 4-1 by B2,XREAL_1:9;
  then
R4: i0-'1 <= 3 by XREAL_1:233,B2;
  per cases;
  suppose
A2: j <> j0;
    now per cases by A2,XXREAL_0:1;
    suppose
      j < j0;
      then (j-'1) < (j0-'1) by XREAL_1:14,P1,P3;
      then (j-'1)+1 <= (j0-'1) by NAT_1:13;
      then
A12:  ((j-'1)+1)*32 <= (j0-'1)*32 by XREAL_1:64;
      (i-'1)*8 <= 3*8 by R2,XREAL_1:64;
      then 8+(i-'1)*8 <= 8+24 by XREAL_1:6;
      then 8+(i-'1)*8+(j-'1)*32 <= 32+(j-'1)*32 by XREAL_1:6;
      then
A13:  8+(i-'1)*8+(j-'1)*32 <= (j0-'1)*32 by A12,XXREAL_0:2;
      0+(j0-'1)*32 <= (i0-'1)*8+(j0-'1)*32 by XREAL_1:6;
      then (j0-'1)*32+0 < (i0-'1)*8+(j0-'1)*32+1 by XREAL_1:8;
      then
A14:  8+(i-'1)*8+(j-'1)*32 < 1+(i0-'1)*8+(j0-'1)*32 by A13,XXREAL_0:2;
      thus A /\ B = {}
      proof
        assume A /\ B <> {};
        then consider x be object such that
A150:   x in A /\ B by XBOOLE_0:def 1;
A15:    x in A & x in B by XBOOLE_0:def 4,A150;
        consider k1 be Nat such that
A16:    x = k1 & 1+(i-'1)*8+(j-'1)*32 <= k1 & k1 <= 8+(i-'1)*8+(j-'1)*32
        by A15;
        consider k2 be Nat such that
A17:    x = k2 & 1+(i0-'1)*8+(j0-'1)*32 <= k2 & k2 <= 8+(i0-'1)*8+(j0-'1)*32
        by A15;
        reconsider x as Nat by A16;
        thus contradiction by A17,A14,XXREAL_0:2,A16;
      end;
    end;
    suppose
      j0 < j;
      then (j0-'1) < (j-'1) by XREAL_1:14,P1,P3;
      then (j0-'1)+1 <= (j-'1) by NAT_1:13;
      then
A12:  ((j0-'1)+1)*32 <= (j-'1)*32 by XREAL_1:64;
      (i0-'1)*8 <= 3*8 by R4,XREAL_1:64;
      then 8+(i0-'1)*8 <= 8+24 by XREAL_1:6;
      then 8+(i0-'1)*8+(j0-'1)*32 <= 32+(j0-'1)*32 by XREAL_1:6;
      then
A13:  8+(i0-'1)*8+(j0-'1)*32 <= (j-'1)*32 by A12,XXREAL_0:2;
      0+(j-'1)*32 <= (i-'1)*8+(j-'1)*32 by XREAL_1:6;
      then (j-'1)*32+0 < (i-'1)*8+(j-'1)*32+1 by XREAL_1:8;
      then
A14:  8+(i0-'1)*8+(j0-'1)*32 < 1+(i-'1)*8+(j-'1)*32 by A13,XXREAL_0:2;
      thus A /\ B = {}
      proof
        assume A /\ B <> {};
        then consider x be object such that
A150:   x in A /\ B by XBOOLE_0:def 1;
A15:    x in A & x in B by XBOOLE_0:def 4,A150;
        consider k1 be Nat such that
A16:    x = k1 & 1+(i-'1)*8+(j-'1)*32 <= k1 & k1 <= 8+(i-'1)*8+(j-'1)*32
        by A15;
        consider k2 be Nat such that
A17:    x = k2 & 1+(i0-'1)*8+(j0-'1)*32 <= k2 & k2 <= 8+(i0-'1)*8+(j0-'1)*32
        by A15;
        reconsider x as Nat by A16;
        thus contradiction by A16,A14,XXREAL_0:2,A17;
      end;
    end;
  end;
  hence A /\ B = {};
  end;
  suppose
A2: j = j0;
    now per cases by A2,AS,XXREAL_0:1;
      suppose i < i0;
      then (i-'1) < (i0-'1) by XREAL_1:14,P2,P4;
      then (i-'1)+1 <= (i0-'1) by NAT_1:13;
      then ((i-'1)+1)*8 <= (i0-'1)*8 by XREAL_1:64;
      then
A13:  (i-'1)*8+8+(j-'1)*32 <= (i0-'1)*8+(j0-'1)*32 by A2,XREAL_1:6;
      (i0-'1)*8+(j0-'1)*32+0 < (i0-'1)*8+(j0-'1)*32+1 by XREAL_1:8;
      then
A14:  8+(i-'1)*8+(j-'1)*32 < 1+(i0-'1)*8+(j0-'1)*32 by A13,XXREAL_0:2;
      thus A /\ B = {}
      proof
        assume A /\ B <> {};
        then consider x be object such that
A150:   x in A /\ B by XBOOLE_0:def 1;
A15:    x in A & x in B by XBOOLE_0:def 4,A150;
        consider k1 be Nat such that
A16:    x = k1 & 1+(i-'1)*8+(j-'1)*32 <= k1 & k1 <= 8+(i-'1)*8+(j-'1)*32
        by A15;
        consider k2 be Nat such that
A17:    x = k2 & 1+(i0-'1)*8+(j0-'1)*32 <= k2 & k2 <= 8+(i0-'1)*8+(j0-'1)*32
        by A15;
        reconsider x as Nat by A16;
        thus contradiction by A16,A17,A14,XXREAL_0:2;
      end;
    end;
    suppose
      i0 < i;
      then (i0-'1) < (i-'1) by XREAL_1:14,P2,P4;
      then (i0-'1)+1 <= (i-'1) by NAT_1:13;
      then ((i0-'1)+1)*8 <= (i-'1)*8 by XREAL_1:64;
      then
A13:  (i0-'1)*8+8+(j0-'1)*32 <= (i-'1)*8+(j-'1)*32 by A2,XREAL_1:6;
      (i-'1)*8+(j-'1)*32+0 < (i-'1)*8+(j-'1)*32+1 by XREAL_1:8;
      then
A14:  8+(i0-'1)*8+(j0-'1)*32 < 1+(i-'1)*8+(j-'1)*32 by A13,XXREAL_0:2;
      thus A /\ B = {}
      proof
        assume A /\ B <> {};
        then consider x be object such that
A150:   x in A /\ B by XBOOLE_0:def 1;
A15:    x in A & x in B by XBOOLE_0:def 4,A150;
        consider k1 be Nat such that
A16:    x = k1 & 1+(i-'1)*8+(j-'1)*32 <= k1 & k1 <= 8+(i-'1)*8+(j-'1)*32
        by A15;
        consider k2 be Nat such that
A17:    x = k2 & 1+(i0-'1)*8+(j0-'1)*32 <= k2 & k2 <= 8+(i0-'1)*8+(j0-'1)*32
        by A15;
        reconsider x as Nat by A16;
        thus contradiction by A16,A14,XXREAL_0:2,A17;
      end;
    end;
  end;
  hence A /\ B = {};
end;
end;
