reserve p,q for Point of TOP-REAL 2,
  i,i1,i2,j,j1,j2,k for Nat,
  r,s for Real,
  G for Matrix of TOP-REAL 2;
reserve f for standard special_circular_sequence;

theorem Th25:
  for G being Go-board holds i < len G & 1 <= j & j < width G
  implies cell(G,i,j) /\ cell(G,i+1,j) = LSeg(G*(i+1,j),G*(i+1,j+1))
proof
  let G be Go-board;
  assume that
A1: i < len G and
A2: 1 <= j and
A3: j < width G;
A4: 0+1 <= i+1 by XREAL_1:6;
A5: i+1 <= len G by A1,NAT_1:13;
  thus cell(G,i,j) /\ cell(G,i+1,j) c= LSeg(G*(i+1,j),G*(i+1,j+1))
  proof
    let x be object;
A6: cell(G,i,j) /\ cell(G,i+1,j)
    = v_strip(G,i) /\ (v_strip(G,i+1) /\ h_strip(G,j) /\
    h_strip(G,j)) by XBOOLE_1:16
      .= v_strip(G,i) /\ (v_strip(G,i+1) /\ (h_strip(G,j) /\ h_strip(G,j)))
    by XBOOLE_1:16
      .= v_strip(G,i) /\ v_strip(G,i+1) /\ h_strip(G,j) by XBOOLE_1:16;
    assume
A7: x in cell(G,i,j) /\ cell(G,i+1,j);
    then
A8: x in v_strip(G,i) /\ v_strip(G,i+1) by A6,XBOOLE_0:def 4;
A9: x in h_strip(G,j) by A6,A7,XBOOLE_0:def 4;
A10: j < j+1 by NAT_1:13;
A11: j+1 <= width G by A3,NAT_1:13;
    then
A12: G*(i+1,j)`2 < G*(i+1,j+1)`2 by A2,A4,A5,A10,Th4;
A13: G*(i+1,j) = |[G*(i+1,j)`1,G*(i+1,j)`2]| by EUCLID:53;
A14: j+1 >= 1 by NAT_1:11;
    G*(i+1,j)`1 = G*(i+1,1)`1 by A2,A3,A4,A5,Th2
      .= G*(i+1,j+1)`1 by A4,A5,A11,A14,Th2;
    then
A15: G*(i+1,j+1) = |[G*(i+1,j)`1,G*(i+1,j+1)`2]| by EUCLID:53;
    reconsider p = x as Point of TOP-REAL 2 by A7;
    i+1 <= len G by A1,NAT_1:13;
    then p in { q : q`1 = G*(i+1,1)`1 } by A8,Th23;
    then ex q st q = p & q`1 = G*(i+1,1)`1;
    then
A16: p`1 = G*(i+1,j)`1 by A2,A3,A4,A5,Th2;
    p in { |[r,s]| : G*(i+1,j)`2 <= s & s <= G*(i+1,j+1)`2 } by A2,A3,A4,A5,A9
,Th5;
    then
A17: ex r,s st ( p = |[r,s]|)&( G*(i+1,j)`2 <= s)&( s <= G*(i+1, j+1)`2);
    then
A18: G*(i+1,j)`2 <= p`2 by EUCLID:52;
    p`2 <= G*(i+1,j+1)`2 by A17,EUCLID:52;
    then p in
    { q : q`1 = G*(i+1,j)`1 & G*(i+1,j)`2 <= q`2 & q`2 <= G*(i+1,j+1)`2 }
    by A16,A18;
    hence thesis by A12,A13,A15,TOPREAL3:9;
  end;
A19: LSeg(G*(i+1,j),G*(i+1,j+1)) c= cell(G,i,j) by A1,A2,A3,Th18;
  LSeg(G*(i+1,j),G*(i+1,j+1)) c= cell(G,i+1,j) by A2,A3,A4,A5,Th19;
  hence thesis by A19,XBOOLE_1:19;
end;
