reserve f for non empty FinSequence of TOP-REAL 2,
  i,j,k,k1,k2,n,i1,i2,j1,j2 for Nat,
  r,s,r1,r2 for Real,
  p,q,p1,q1 for Point of TOP-REAL 2,
  G for Go-board;

theorem Th21:
  for i1,j1,i2,j2 being Nat st 1 <= i1 & i1 <= len G &
1 <= j1 & j1+1 <= width G & 1 <= i2 & i2+1 <= len G & 1 <= j2 & j2 <= width G &
LSeg(G*(i1,j1),G*(i1,j1+1)) meets LSeg(G*(i2,j2),G*(i2+1,j2)) holds (i1 = i2 or
  i1 = i2+1) & (j1 = j2 or j1+1 = j2)
proof
  let i1,j1,i2,j2 be Nat such that
A1: 1 <= i1 and
A2: i1 <= len G and
A3: 1 <= j1 and
A4: j1+1 <= width G and
A5: 1 <= i2 and
A6: i2+1 <= len G and
A7: 1 <= j2 and
A8: j2 <= width G;
  assume LSeg(G*(i1,j1),G*(i1,j1+1)) meets LSeg(G*(i2,j2),G*(i2+1,j2));
  then consider x being object such that
A9: x in LSeg(G*(i1,j1),G*(i1,j1+1)) and
A10: x in LSeg(G*(i2,j2),G*(i2+1,j2)) by XBOOLE_0:3;
  reconsider p = x as Point of TOP-REAL 2 by A9;
  consider r1 such that
A11: p = (1-r1)*(G*(i1,j1))+r1*(G*(i1,j1+1)) and
A12: r1 >= 0 and
A13: r1 <= 1 by A9;
  consider r2 such that
A14: p = (1-r2)*(G*(i2,j2))+r2*(G*(i2+1,j2)) and
A15: r2 >= 0 and
A16: r2 <= 1 by A10;
A17: i2 < len G by A6,NAT_1:13;
A18: 1 <= j1+1 by A3,NAT_1:13;
  then
A19: G*(i1,j1+1)`2 = G*(1,j1+1)`2 by A1,A2,A4,GOBOARD5:1
    .= G*(i2,j1+1)`2 by A4,A5,A18,A17,GOBOARD5:1;
A20: j1 < width G by A4,NAT_1:13;
  then
A21: G*(i1,j1)`2 = G*(1,j1)`2 by A1,A2,A3,GOBOARD5:1
    .= G*(i2,j1)`2 by A3,A5,A20,A17,GOBOARD5:1;
A22: (1-r2)*(G*(i2,j2))`2+r2*(G*(i2+1,j2))`2 = ((1-r2)*(G*(i2,j2)))`2+r2*(G*
  (i2+1,j2))`2 by TOPREAL3:4
    .= ((1-r2)*(G*(i2,j2)))`2+(r2*(G*(i2+1,j2)))`2 by TOPREAL3:4
    .= p`2 by A14,TOPREAL3:2
    .= ((1-r1)*(G*(i1,j1)))`2+(r1*(G*(i1,j1+1)))`2 by A11,TOPREAL3:2
    .= (1-r1)*(G*(i1,j1))`2+(r1*(G*(i1,j1+1)))`2 by TOPREAL3:4
    .= (1-r1)*(G*(i2,j1))`2+r1*(G*(i2,j1+1))`2 by A19,A21,TOPREAL3:4;
A23: 1 <= i2+1 by A5,NAT_1:13;
  thus i1 = i2 or i1 = i2+1
  proof
A24: G*(i2,j2)`1 = G*(i2,1)`1 by A5,A7,A8,A17,GOBOARD5:2
      .= G*(i2,j1)`1 by A3,A5,A20,A17,GOBOARD5:2;
A25: G*(i2+1,j2)`1 = G*(i2+1,1)`1 by A6,A7,A8,A23,GOBOARD5:2
      .= G*(i2+1,j1)`1 by A3,A6,A20,A23,GOBOARD5:2;
A26: (1-r1)*(G*(i1,j1))`1+r1*(G*(i1,j1+1))`1 = ((1-r1)*(G*(i1,j1)))`1+r1*(
    G*(i1,j1+1))`1 by TOPREAL3:4
      .= ((1-r1)*(G*(i1,j1)))`1+(r1*(G*(i1,j1+1)))`1 by TOPREAL3:4
      .= p`1 by A11,TOPREAL3:2
      .= ((1-r2)*(G*(i2,j2)))`1+(r2*(G*(i2+1,j2)))`1 by A14,TOPREAL3:2
      .= (1-r2)*(G*(i2,j2))`1+(r2*(G*(i2+1,j2)))`1 by TOPREAL3:4
      .= (1-r2)*(G*(i2,j1))`1+r2*(G*(i2+1,j1))`1 by A25,A24,TOPREAL3:4;
A27: G*(i1,j1)`1 = G*(i1,1)`1 by A1,A2,A3,A20,GOBOARD5:2
      .= G*(i1,j1+1)`1 by A1,A2,A4,A18,GOBOARD5:2;
    assume
A28: not thesis;
    per cases by A28,XXREAL_0:1;
    suppose
A29:  i1 < i2 & i1 < i2+1;
      i2 < i2+1 by XREAL_1:29;
      then G*(i2,j1)`1 < G*(i2+1,j1)`1 by A3,A5,A6,A20,GOBOARD5:3;
      then
A30:  (1-r2)*(G*(i2,j1))`1+r2*(G*(i2,j1))`1 = 1*(G*(i2,j1))`1 & r2*(G*(i2
      ,j1))`1 <= r2*(G*(i2+1,j1))`1 by A15,XREAL_1:64;
      G*(i1,j1)`1 < G*(i2,j1)`1 by A1,A3,A20,A17,A29,GOBOARD5:3;
      hence contradiction by A26,A27,A30,XREAL_1:6;
    end;
    suppose
      i1 < i2 & i2+1 < i1;
      hence thesis by NAT_1:13;
    end;
    suppose
      i2 < i1 & i1 < i2+1;
      hence thesis by NAT_1:13;
    end;
    suppose
A31:  i2+1 < i1;
      i2 < i2 + 1 by XREAL_1:29;
      then
A32:  G*(i2,j1)`1 <= G*(i2+1,j1)`1 by A3,A5,A6,A20,GOBOARD5:3;
      1-r2 >= 0 by A16,XREAL_1:48;
      then
      (1-r2)*(G*(i2+1,j1))`1+r2*(G*(i2+1,j1))`1 = 1*(G*(i2+1,j1))`1 & (1-
      r2)*(G*( i2,j1))`1 <= (1-r2)*(G*(i2+1,j1))`1 by A32,XREAL_1:64;
      then (1-r2)*(G*(i2,j1))`1+r2*(G*(i2+1,j1))`1 <= G*(i2+1,j1)`1 by
XREAL_1:6;
      hence contradiction by A2,A3,A20,A23,A26,A27,A31,GOBOARD5:3;
    end;
  end;
A33: G*(i2,j2)`2 = G*(1,j2)`2 by A5,A7,A8,A17,GOBOARD5:1
    .= G*(i2+1,j2)`2 by A6,A7,A8,A23,GOBOARD5:1;
  assume
A34: not thesis;
  per cases by A34,XXREAL_0:1;
  suppose
A35: j2 < j1 & j2 < j1+1;
    j1 < j1+1 by XREAL_1:29;
    then G*(i2,j1)`2 < G*(i2,j1+1)`2 by A3,A4,A5,A17,GOBOARD5:4;
    then
A36: (1-r1)*(G*(i2,j1))`2+r1*(G*(i2,j1))`2 = 1*(G*(i2,j1))`2 & r1*(G*(i2,
    j1))`2 <= r1*(G*(i2,j1+1))`2 by A12,XREAL_1:64;
    G*(i2,j2)`2 < G*(i2,j1)`2 by A5,A7,A20,A17,A35,GOBOARD5:4;
    hence contradiction by A22,A33,A36,XREAL_1:6;
  end;
  suppose
    j2 < j1 & j1+1 < j2;
    hence thesis by NAT_1:13;
  end;
  suppose
    j1 < j2 & j2 < j1+1;
    hence thesis by NAT_1:13;
  end;
  suppose
A37: j1+1 < j2;
    j1 < j1 + 1 by XREAL_1:29;
    then
A38: G*(i2,j1)`2 <= G*(i2,j1+1)`2 by A3,A4,A5,A17,GOBOARD5:4;
    1-r1 >= 0 by A13,XREAL_1:48;
    then
    (1-r1)*(G*(i2,j1+1))`2+r1*(G*(i2,j1+1))`2 = 1*(G*(i2,j1+1))`2 & (1-r1
    )*(G*( i2,j1))`2 <= (1-r1)*(G*(i2,j1+1))`2 by A38,XREAL_1:64;
    then (1-r1)*(G*(i2,j1))`2+r1*(G*(i2,j1+1))`2 <= G* (i2,j1+1)`2 by XREAL_1:6
;
    hence contradiction by A5,A8,A18,A17,A22,A33,A37,GOBOARD5:4;
  end;
end;
