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 Th26:
  1 <= i1 & i1+1 <= len G & 1 <= j1 & j1 <= width G & 1 <= i2 & i2
+1 <= len G & 1 <= j2 & j2 <= width G & 1/2*(G*(i1,j1)+G*(i1+1,j1)) in LSeg(G*(
  i2,j2),G*(i2+1,j2)) implies i1 = i2 & j1 = j2
proof
  assume that
A1: 1 <= i1 and
A2: i1+1 <= len G and
A3: 1 <= j1 & j1 <= width G and
A4: 1 <= i2 and
A5: i2+1 <= len G and
A6: 1 <= j2 & j2 <= width G;
  set mi = 1/2*(G*(i1,j1)+G*(i1+1,j1));
A7: 1/2*(G*(i1,j1))+1/2*(G*(i1+1,j1)) = 1/2*(G*(i1,j1)+G*(i1+1,j1)) by
RLVECT_1:def 5;
  then
A8: mi in LSeg(G*(i1,j1),G*(i1+1,j1)) by Lm1;
  assume
A9: mi in LSeg(G*(i2,j2),G*(i2+1,j2));
  then
A10: LSeg(G*(i1,j1),G*(i1+1,j1)) meets LSeg(G*(i2,j2),G*(i2+1,j2)) by A8,
XBOOLE_0:3;
  then
A11: j1 = j2 by A1,A2,A3,A4,A5,A6,Th20;
  now
    i1 < i1+1 by XREAL_1:29;
    then
A12: G*(i1+1,j1)`1 > G*(i1,j1)`1 by A1,A2,A3,GOBOARD5:3;
    assume
A13: |.i1-i2.| = 1;
    per cases by A13,SEQM_3:41;
    suppose
A14:  i1 = i2+1;
      then
      LSeg(G*(i2,j2),G*(i2+1,j2)) /\ LSeg(G*(i2+1,j2),G*(i2+2,j2)) = { G*
      (i2+1,j2) } by A2,A4,A6,Th14;
      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+1,j1)) = 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+1,j1)) by Th3;
      hence contradiction by A12,RLVECT_1:36;
    end;
    suppose
A15:  i1+1 = i2;
      then
      LSeg(G*(i1,j2),G*(i1+1,j2)) /\ LSeg(G*(i1+1,j2),G*(i1+2,j2)) = { G*
      (i1+1,j2) } by A1,A5,A6,Th14;
      then mi in { G*(i2,j1) } by A9,A8,A11,A15,XBOOLE_0:def 4;
      then 1/2*(G*(i1,j1))+1/2*(G*(i1+1,j1)) = G*(i2,j1) by A7,TARSKI:def 1
        .= (1/2+1/2)*(G*(i2,j1)) by RLVECT_1:def 8
        .= 1/2*(G*(i2,j1))+1/2*(G*(i2,j1)) by RLVECT_1:def 6;
      then 1/2*(G*(i1,j1)) = 1/2*(G*(i1+1,j1)) by A15,Th3;
      hence contradiction by A12,RLVECT_1:36;
    end;
  end;
  then |.i1-i2.| = 0 by A1,A2,A3,A4,A5,A6,A10,Th20,NAT_1:25;
  hence i1 = i2 by Th2;
  thus thesis by A1,A2,A3,A4,A5,A6,A10,Th20;
end;
