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 Th27:
  1 <= i1 & i1+1 <= len G & 1 <= j1 & j1 <= width G implies not ex
i2,j2 st 1 <= i2 & i2 <= len G & 1 <= j2 & j2+1 <= width G & 1/2*(G*(i1,j1)+G*(
  i1+1,j1)) in LSeg(G*(i2,j2),G*(i2,j2+1))
proof
  assume that
A1: 1 <= i1 & i1+1 <= len G and
A2: 1 <= j1 & j1 <= width G;
A3: i1 < i1+1 by XREAL_1:29;
  set mi = 1/2*(G*(i1,j1)+G*(i1+1,j1));
  given i2,j2 such that
A4: 1 <= i2 & i2 <= len G and
A5: 1 <= j2 & j2+1 <= width G and
A6: mi in LSeg(G*(i2,j2),G*(i2,j2+1));
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;
  then
A9: LSeg(G*(i1,j1),G*(i1+1,j1)) meets LSeg(G*(i2,j2),G*(i2,j2+1)) by A6,
XBOOLE_0:3;
  per cases by A1,A2,A4,A5,A9,Th21;
  suppose
A10: j1 = j2 & i1 = i2;
    then
    LSeg(G*(i1,j1),G*(i1+1,j1)) /\ LSeg(G*(i1,j1+1),G*(i1,j1)) = { G*(i1,
    j1) } by A1,A5,Th17;
    then mi in { G*(i1,j1) } by A6,A8,A10,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
A11: 1/2*(G*(i1+1,j1)) = 1/2*(G*(i1,j1)) by Th3;
    G*(i1+1,j1)`1 > G*(i1,j1)`1 by A1,A2,A3,GOBOARD5:3;
    hence contradiction by A11,RLVECT_1:36;
  end;
  suppose
A12: j1 = j2 & i1+1 = i2;
    then
    LSeg(G*(i1,j1),G*(i1+1,j1)) /\ LSeg(G*(i1+1,j1+1),G*(i1+1,j1)) = { G*
    (i1+1,j1) } by A1,A5,Th18;
    then mi in { G*(i1+1,j1) } by A6,A8,A12,XBOOLE_0:def 4;
    then 1/2*(G*(i1,j1))+1/2*(G*(i1+1,j1)) = G*(i1+1,j1) by A7,TARSKI:def 1
      .= (1/2+1/2)*(G*(i1+1,j1)) by RLVECT_1:def 8
      .= 1/2*(G*(i1+1,j1))+1/2*(G*(i1+1,j1)) by RLVECT_1:def 6;
    then
A13: 1/2*(G*(i1+1,j1)) = 1/2*(G*(i1,j1)) by Th3;
    G*(i1+1,j1)`1 > G*(i1,j1)`1 by A1,A2,A3,GOBOARD5:3;
    hence contradiction by A13,RLVECT_1:36;
  end;
  suppose
A14: j1 = j2+1 & i1 = i2;
    then
    LSeg(G*(i1,j1),G*(i1+1,j1)) /\ LSeg(G*(i1,j1),G*(i1,j2)) = { G*(i1,j1
    ) } by A1,A5,Th15;
    then mi in { G*(i1,j1) } by A6,A8,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
A15: 1/2*(G*(i1+1,j1)) = 1/2*(G*(i1,j1)) by Th3;
    G*(i1+1,j1)`1 > G*(i1,j1)`1 by A1,A2,A3,GOBOARD5:3;
    hence contradiction by A15,RLVECT_1:36;
  end;
  suppose
A16: j1 = j2+1 & i1+1 = i2;
    then LSeg(G*(i1,j1),G*(i1+1,j1)) /\ LSeg(G*(i1+1,j1),G*(i1+1,j2)) = { G*(
    i1+1,j1) } by A1,A5,Th16;
    then mi in { G*(i1+1,j1) } by A6,A8,A16,XBOOLE_0:def 4;
    then 1/2*(G*(i1,j1))+1/2*(G*(i1+1,j1)) = G*(i1+1,j1) by A7,TARSKI:def 1
      .= (1/2+1/2)*(G*(i1+1,j1)) by RLVECT_1:def 8
      .= 1/2*(G*(i1+1,j1))+1/2*(G*(i1+1,j1)) by RLVECT_1:def 6;
    then
A17: 1/2*(G*(i1+1,j1)) = 1/2*(G*(i1,j1)) by Th3;
    G*(i1+1,j1)`1 > G*(i1,j1)`1 by A1,A2,A3,GOBOARD5:3;
    hence contradiction by A17,RLVECT_1:36;
  end;
end;
