reserve i,i1,i2,i9,i19,j,j1,j2,j9,j19,k,l,m,n for Nat;
reserve r,s,r9,s9 for Real;
reserve D for set,
  f for FinSequence of D,
  G for Matrix of D;
reserve G for Go-board,
  p for Point of TOP-REAL 2;

theorem
  i+k < len G & 1 <= j & j < width G & cell(G,i,j) meets cell(G,i+k,j)
  implies k <= 1
proof
  assume that
A1: i+k < len G and
A2: 1 <= j and
A3: j < width G and
A4: cell(G,i,j) meets cell(G,i+k,j) and
A5: k > 1;
  cell(G,i,j) /\ cell(G,i+k,j) <> {} by A4;
  then consider p such that
A6: p in cell(G,i,j) /\ cell(G,i+k,j) by SUBSET_1:4;
A7: p in cell(G,i,j) by A6,XBOOLE_0:def 4;
A8: p in cell(G,i+k,j) by A6,XBOOLE_0:def 4;
  i+k >= 1 by A5,NAT_1:12;
  then cell(G,i+k,j) = { |[r9,s9]| : G*(i+k,1)`1 <= r9 & r9 <= G*(i+k+1,1)`1 &
  G*(1,j)`2 <= s9 & s9 <= G*(1,j+1)`2 } by A1,A2,A3,GOBRD11:32;
  then consider r9,s9 such that
A9: p = |[r9,s9]| and
A10: G*(i+k,1)`1 <= r9 and r9 <= G*(i+k+1,1)`1
  and G*(1,j)`2 <= s9
  and s9 <= G*(1,j+1)`2 by A8;
A11: 1 < width G by A2,A3,XXREAL_0:2;
A12: i = 0 or i >= 1+0 by NAT_1:9;
  per cases by A12;
  suppose
A13: i = 0;
    then cell(G,i,j) = {|[r,s]| : r <= G*(1,1)`1 & G*(1,j)`2 <= s & s <= G*
    (1,j+1)`2} by A2,A3,GOBRD11:26;
    then consider r,s such that
A14: p = |[r,s]| and
A15: r <= G*(1,1)`1 and G*(1,j)`2 <= s
    and s <= G*(1,j+1)`2 by A7;
A16: k <= len G by A1,NAT_1:12;
A17: p`1 = r by A14,EUCLID:52;
    p`1 = r9 by A9,EUCLID:52;
    then G*(k,1)`1 <= G*(1,1)`1 by A10,A13,A15,A17,XXREAL_0:2;
    hence contradiction by A5,A11,A16,GOBOARD5:3;
  end;
  suppose
A18: i >= 1;
    i < len G by A1,NAT_1:12;
    then cell(G,i,j) =
    {|[r,s]|: G*(i,1)`1 <= r & r <= G*(i+1,1)`1 & G*(1,j)`2 <= s & s <=
    G*(1,j+1)`2} by A2,A3,A18,GOBRD11:32;
    then consider r,s such that
A19: p = |[r,s]| and G*(i,1)`1 <= r and
A20: r <= G*(i+1,1)`1 and G*(1,j)`2 <= s
    and s <= G*(1,j+1)`2 by A7;
A21: 1 <= i+1 by NAT_1:12;
A22: i+1 < k+i by A5,XREAL_1:6;
A23: p`1 = r by A19,EUCLID:52;
    p`1 = r9 by A9,EUCLID:52;
    then G*(i+k,1)`1 <= G*(i+1,1)`1 by A10,A20,A23,XXREAL_0:2;
    hence contradiction by A1,A11,A21,A22,GOBOARD5:3;
  end;
end;
