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 Th25:
  1 <= i1 & i1 <= len G & 1 <= j1 & j1+1 <= width G & 1 <= i2 & i2
<= len G & 1 <= j2 & j2+1 <= width G & 1/2*(G*(i1,j1)+G*(i1,j1+1)) in LSeg(G*(
  i2,j2),G*(i2,j2+1)) implies i1 = i2 & j1 = j2
proof
  assume 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;
  set mi = 1/2*(G*(i1,j1)+G*(i1,j1+1));
A7: 1/2*(G*(i1,j1))+1/2*(G*(i1,j1+1)) = 1/2*(G*(i1,j1)+G*(i1,j1+1)) by
RLVECT_1:def 5;
  then
A8: mi in LSeg(G*(i1,j1),G*(i1,j1+1)) by Lm1;
  assume
A9: mi in LSeg(G*(i2,j2),G*(i2,j2+1));
  then
A10: LSeg(G*(i1,j1),G*(i1,j1+1)) meets LSeg(G*(i2,j2),G*(i2,j2+1)) by A8,
XBOOLE_0:3;
  hence
A11: i1 = i2 by A1,A2,A3,A4,A5,A6,Th19;
  now
    j1 < j1+1 by XREAL_1:29;
    then
A12: G*(i1,j1+1)`2 > G*(i1,j1)`2 by A1,A2,A3,GOBOARD5:4;
    assume
A13: |.j1-j2.| = 1;
    per cases by A13,SEQM_3:41;
    suppose
A14:  j1 = j2+1;
      then
      LSeg(G*(i2,j2),G*(i2,j2+1)) /\ LSeg(G*(i2,j2+1),G*(i2,j2+2)) = { G*
      (i2,j2+1) } by A3,A4,A5,Th13;
      then mi in { G*(i1,j1) } by A9,A8,A11,A14,XBOOLE_0:def 4;
      then 1/2*(G*(i1,j1))+1/2*(G*(i1,j1+1)) = G*(i1,j1) by A7,TARSKI:def 1
        .= (1/2+1/2)*(G*(i1,j1)) by RLVECT_1:def 8
        .= 1/2*(G*(i1,j1))+1/2*(G*(i1,j1)) by RLVECT_1:def 6;
      then 1/2*(G*(i1,j1)) = 1/2*(G*(i1,j1+1)) by Th3;
      hence contradiction by A12,RLVECT_1:36;
    end;
    suppose
A15:  j1+1 = j2;
      then
      LSeg(G*(i2,j1),G*(i2,j1+1)) /\ LSeg(G*(i2,j1+1),G*(i2,j1+2)) = { G*
      (i2,j1+1) } by A2,A4,A6,Th13;
      then mi in { G*(i1,j2) } by A9,A8,A11,A15,XBOOLE_0:def 4;
      then 1/2*(G*(i1,j1))+1/2*(G*(i1,j1+1)) = G*(i1,j2) by A7,TARSKI:def 1
        .= (1/2+1/2)*(G*(i1,j2)) by RLVECT_1:def 8
        .= 1/2*(G*(i1,j2))+1/2*(G*(i1,j2)) by RLVECT_1:def 6;
      then 1/2*(G*(i1,j1)) = 1/2*(G*(i1,j1+1)) by A15,Th3;
      hence contradiction by A12,RLVECT_1:36;
    end;
  end;
  then |.j1-j2.| = 0 by A1,A2,A3,A4,A5,A6,A10,Th19,NAT_1:25;
  hence thesis by Th2;
end;
