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 <= width GoB f & 1 <= k & k+1 <
len f & LSeg((GoB f)*(i+1,j),(GoB f)*(i+2,j)) = LSeg(f,k) & LSeg((GoB f)*(i,j),
(GoB f)*(i+1,j)) = LSeg(f,k+1) implies f/.k = (GoB f)*(i+2,j) & f/.(k+1) = (GoB
  f)*(i+1,j) & f/.(k+2) = (GoB f)*(i,j)
proof
  assume that
A1: 1 <= i and
A2: i+1 < len GoB f and
A3: 1 <= j & j <= width GoB f and
A4: 1 <= k and
A5: k+1 < len f and
A6: LSeg((GoB f)*(i+1,j),(GoB f)*(i+2,j)) = LSeg(f,k) and
A7: LSeg((GoB f)*(i,j),(GoB f)*(i+1,j)) = LSeg(f,k+1);
A8: i < i+2 by XREAL_1:29;
  i+(1+1) = i+1+1;
  then i+2 <= len GoB f by A2,NAT_1:13;
  then
A9: (GoB f)*(i,j)`1 < (GoB f)*(i+2,j)`1 by A1,A3,A8,GOBOARD5:3;
A10: 1 <= k+1 by NAT_1:11;
A11: k+(1+1) = k+1+1;
  then k+2 <= len f by A5,NAT_1:13;
  then
A12: LSeg((GoB f)*(i,j),(GoB f)*(i+1,j)) = LSeg(f/.(k+1),f/.(k+2)) by A7,A11
,A10,TOPREAL1:def 3;
  then
A13: (GoB f)*(i,j) = f/.(k+2) & (GoB f)*(i+1,j) = f/.(k+1) or (GoB f)*(i,j)
  = f/.(k+1) & (GoB f)*(i+1,j) = f/.(k+2) by SPPOL_1:8;
A14: LSeg((GoB f)*(i+1,j),(GoB f)*(i+2,j)) = LSeg(f/.k,f/.(k+1)) by A4,A5,A6,
TOPREAL1:def 3;
  then
  (GoB f)*(i+1,j) = f/.(k+1) & (GoB f)*(i+2,j) = f/.k or (GoB f)*(i+1,j) =
  f/.k & (GoB f)*(i+2,j) = f/.(k+1) by SPPOL_1:8;
  hence f/.k = (GoB f)*(i+2,j) by A12,A9,SPPOL_1:8;
  thus f/.(k+1) = (GoB f)*(i+1,j) by A14,A13,A9,SPPOL_1:8;
  thus thesis by A14,A13,A9,SPPOL_1:8;
end;
