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;
reserve f for non constant standard special_circular_sequence;

theorem
  1 <= i & i+1 <= len GoB f & 1 <= j & j+1 <= width GoB f & 1 <= k & k+1
< len f & LSeg((GoB f)*(i,j+1),(GoB f)*(i+1,j+1)) = LSeg(f,k) & LSeg((GoB f)*(i
,j),(GoB f)*(i,j+1)) = LSeg(f,k+1) implies f/.k = (GoB f)*(i+1,j+1) & f/.(k+1)
  = (GoB f)*(i,j+1) & f/.(k+2) = (GoB f)*(i,j)
proof
  assume that
A1: 1 <= i and
A2: i+1 <= len GoB f and
A3: 1 <= j and
A4: j+1 <= width GoB f and
A5: 1 <= k and
A6: k+1 < len f and
A7: LSeg((GoB f)*(i,j+1),(GoB f)*(i+1,j+1)) = LSeg(f,k) and
A8: LSeg((GoB f)*(i,j),(GoB f)*(i,j+1)) = LSeg(f,k+1);
A9: j < width GoB f by A4,NAT_1:13;
A10: j < j+1 by NAT_1:13;
A11: 1 <= i+1 by NAT_1:11;
  i < len GoB f by A2,NAT_1:13;
  then (GoB f)*(i,j)`2 = (GoB f)*(1,j)`2 by A1,A3,A9,GOBOARD5:1
    .= (GoB f)*(i+1,j)`2 by A2,A3,A11,A9,GOBOARD5:1;
  then
A12: (GoB f)*(i,j) <> (GoB f)*(i+1,j+1) by A2,A3,A4,A11,A10,GOBOARD5:4;
A13: 1 <= k+1 by NAT_1:11;
A14: k+(1+1) = k+1+1;
  then k+2 <= len f by A6,NAT_1:13;
  then
A15: LSeg((GoB f)*(i,j),(GoB f)*(i,j+1)) = LSeg(f/.(k+1),f/.(k+2)) by A8,A14
,A13,TOPREAL1:def 3;
  then
A16: (GoB f)*(i,j) = f/.(k+2) & (GoB f)*(i,j+1) = f/.(k+1) or (GoB f)*(i,j)
  = f/.(k+1) & (GoB f)*(i,j+1) = f/.(k+2) by SPPOL_1:8;
A17: LSeg((GoB f)*(i,j+1),(GoB f)*(i+1,j+1)) = LSeg(f/.k,f/.(k+1)) by A5,A6,A7,
TOPREAL1:def 3;
  then
  (GoB f)*(i+1,j+1) = f/.(k+1) & (GoB f)*(i,j+1) = f/.k or (GoB f)*(i+1,j
  +1) = f/.k & (GoB f)* (i,j+1) = f/.(k+1) by SPPOL_1:8;
  hence f/.k = (GoB f)*(i+1,j+1) by A15,A12,SPPOL_1:8;
  thus f/.(k+1) = (GoB f)*(i,j+1) by A17,A16,A12,SPPOL_1:8;
  thus thesis by A17,A16,A12,SPPOL_1:8;
end;
