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 Th19:
  for i1,j1,i2,j2 being Nat st 1 <= i1 & i1 <= len G &
1 <= j1 & j1+1 <= width G & 1 <= i2 & i2 <= len G & 1 <= j2 & j2+1 <= width G &
  LSeg(G*(i1,j1),G*(i1,j1+1)) meets LSeg(G*(i2,j2),G*(i2,j2+1)) holds i1 = i2 &
  |.j1-j2.| <= 1
proof
  let i1,j1,i2,j2 be Nat such that
A1: 1 <= i1 & i1 <= len G and
A2: 1 <= j1 and
A3: j1+1 <= width G and
A4: 1 <= i2 & i2 <= len G and
A5: 1 <= j2 and
A6: j2+1 <= width G;
A7: 1 <= j1+1 by A2,NAT_1:13;
A8: j1 < width G by A3,NAT_1:13;
  assume LSeg(G*(i1,j1),G*(i1,j1+1)) meets LSeg(G*(i2,j2),G*(i2,j2+1));
  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,j2+1)) 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,j2+1)) and
A15: r2 >= 0 and
A16: r2 <= 1 by A10;
A17: 1 <= j2+1 by A5,NAT_1:13;
A18: j2 < width G by A6,NAT_1:13;
  assume
A19: not thesis;
  per cases by A19;
  suppose
    i1 <> i2;
    then
A20: i1 < i2 or i2 < i1 by XXREAL_0:1;
A21: G*(i2,j2)`1 = G*(i2,1)`1 by A4,A5,A18,GOBOARD5:2
      .= G*(i2,j2+1)`1 by A4,A6,A17,GOBOARD5:2;
    G*(i1,j1)`1 = G*(i1,1)`1 by A1,A2,A8,GOBOARD5:2
      .= G*(i1,j1+1)`1 by A1,A3,A7,GOBOARD5:2;
    then 1*(G*(i1,j1))`1 = (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,j2+1)))`1 by A14,TOPREAL3:2
      .= (1-r2)*(G*(i2,j2))`1+(r2*(G*(i2,j2+1)))`1 by TOPREAL3:4
      .= (1-r2)*(G*(i2,j2))`1+r2*(G*(i2,j2+1))`1 by TOPREAL3:4
      .= G*(i2,1)`1 by A4,A6,A17,A21,GOBOARD5:2
      .= G*(i2,j1)`1 by A2,A4,A8,GOBOARD5:2;
    hence contradiction by A1,A2,A4,A8,A20,GOBOARD5:3;
  end;
  suppose
A22: |.j1-j2.| > 1;
A23: G*(i2,j2+1)`2 = G*(1,j2+1)`2 by A4,A6,A17,GOBOARD5:1
      .= G*(i1,j2+1)`2 by A1,A6,A17,GOBOARD5:1;
A24: G*(i2,j2)`2 = G*(1,j2)`2 by A4,A5,A18,GOBOARD5:1
      .= G*(i1,j2)`2 by A1,A5,A18,GOBOARD5:1;
A25: (1-r1)*(G*(i1,j1))`2+r1*(G*(i1,j1+1))`2 = ((1-r1)*(G*(i1,j1)))`2+r1*(
    G*(i1,j1+1))`2 by TOPREAL3:4
      .= ((1-r1)*(G*(i1,j1)))`2+(r1*(G*(i1,j1+1)))`2 by TOPREAL3:4
      .= p`2 by A11,TOPREAL3:2
      .= ((1-r2)*(G*(i2,j2)))`2+(r2*(G*(i2,j2+1)))`2 by A14,TOPREAL3:2
      .= (1-r2)*(G*(i2,j2))`2+(r2*(G*(i2,j2+1)))`2 by TOPREAL3:4
      .= (1-r2)*(G*(i1,j2))`2+r2*(G*(i1,j2+1))`2 by A23,A24,TOPREAL3:4;
    now
      per cases by A22,Th1;
      suppose
A26:    j1+1 < j2;
        j2 < j2+1 by XREAL_1:29;
        then G*(i1,j2)`2 < G*(i1,j2+1)`2 by A1,A5,A6,GOBOARD5:4;
        then
        (1-r2)*(G*(i1,j2))`2+r2*(G*(i1,j2))`2 = 1*(G*(i1,j2))`2 & r2*(G*(
        i1,j2))`2 <= r2*(G*(i1,j2+1))`2 by A15,XREAL_1:64;
        then
A27:    G*(i1,j2)`2 <= (1-r2)*(G*(i1,j2))`2+r2*(G*(i1,j2+1))`2 by XREAL_1:6;
        j1 < j1 + 1 by XREAL_1:29;
        then
A28:    G*(i1,j1)`2 <= G*(i1,j1+1)`2 by A1,A2,A3,GOBOARD5:4;
        1-r1 >= 0 by A13,XREAL_1:48;
        then
        (1-r1)*(G*(i1,j1+1))`2+r1*(G*(i1,j1+1))`2 = 1*(G*(i1,j1+1))`2 & (
        1-r1)*(G*( i1,j1))`2 <= (1-r1)*(G*(i1,j1+1))`2 by A28,XREAL_1:64;
        then
A29:    (1-r1)*(G*(i1,j1))`2+r1*(G*(i1,j1+1))`2 <= G*(i1,j1+1)`2 by XREAL_1:6;
        G*(i1,j1+1)`2 < G*(i1,j2)`2 by A1,A7,A18,A26,GOBOARD5:4;
        hence contradiction by A25,A29,A27,XXREAL_0:2;
      end;
      suppose
A30:    j2+1 < j1;
        j1 < j1+1 by XREAL_1:29;
        then G*(i1,j1)`2 < G*(i1,j1+1)`2 by A1,A2,A3,GOBOARD5:4;
        then
        (1-r1)*(G*(i1,j1))`2+r1*(G*(i1,j1))`2 = 1*(G*(i1,j1))`2 & r1*(G*(
        i1,j1))`2 <= r1*(G*(i1,j1+1))`2 by A12,XREAL_1:64;
        then
A31:    G*(i1,j1)`2 <= (1-r1)*(G*(i1,j1))`2+r1*(G* (i1,j1+1))`2 by XREAL_1:6;
        j2 < j2 + 1 by XREAL_1:29;
        then
A32:    G*(i1,j2)`2 <= G*(i1,j2+1)`2 by A1,A5,A6,GOBOARD5:4;
        1-r2 >= 0 by A16,XREAL_1:48;
        then
        (1-r2)*(G*(i1,j2+1))`2+r2*(G*(i1,j2+1))`2 = 1*(G*(i1,j2+1))`2 & (
        1-r2)*(G*( i1,j2))`2 <= (1-r2)*(G*(i1,j2+1))`2 by A32,XREAL_1:64;
        then
A33:    (1-r2)*(G*(i1,j2))`2+r2*(G*(i1,j2+1))`2 <= G*(i1,j2+1)`2 by XREAL_1:6;
        G*(i1,j2+1)`2 < G*(i1,j1)`2 by A1,A8,A17,A30,GOBOARD5:4;
        hence contradiction by A25,A33,A31,XXREAL_0:2;
      end;
    end;
    hence contradiction;
  end;
end;
