reserve G for Go-board,
  i,j,k,m,n for Nat;

theorem Th32:
  for f being non empty s.n.c. unfolded FinSequence of TOP-REAL 2,
  i st 1 <= i & i < len f holds LSeg(f,i) /\ rng f = {f/.i,f/.(i+1)}
proof
  let f be non empty s.n.c. unfolded FinSequence of TOP-REAL 2, i such that
A1: 1 <= i and
A2: i < len f;
A3: i in dom f by A1,A2,FINSEQ_3:25;
  then f/.i = f.i by PARTFUN1:def 6;
  then
A4: f/.i in rng f by A3,FUNCT_1:3;
  assume
A5: LSeg(f,i) /\ rng f <> {f/.i,f/.(i+1)};
A6: i+1 <= len f by A2,NAT_1:13;
  then f/.i in LSeg(f,i) by A1,TOPREAL1:21;
  then
A7: f/.i in LSeg(f,i) /\ rng f by A4,XBOOLE_0:def 4;
A8: 1 < i+1 by A1,XREAL_1:29;
  then
A9: i+1 in dom f by A6,FINSEQ_3:25;
  then f/.(i+1) = f.(i+1) by PARTFUN1:def 6;
  then
A10: f/.(i+1) in rng f by A9,FUNCT_1:3;
  f/.(i+1) in LSeg(f,i) by A1,A6,TOPREAL1:21;
  then f/.(i+1) in LSeg(f,i) /\ rng f by A10,XBOOLE_0:def 4;
  then {f/.i,f/.(i+1)} c= LSeg(f,i) /\ rng f by A7,ZFMISC_1:32;
  then not LSeg(f,i) /\ rng f c= {f/.i,f/.(i+1)} by A5,XBOOLE_0:def 10;
  then consider x being object such that
A11: x in LSeg(f,i) /\ rng f and
A12: not x in {f/.i,f/.(i+1)};
  reconsider x as Point of TOP-REAL 2 by A11;
A13: x in LSeg(f,i) by A11,XBOOLE_0:def 4;
  x in rng f by A11,XBOOLE_0:def 4;
  then consider j being object such that
A14: j in dom f and
A15: x = f.j by FUNCT_1:def 3;
A16: x = f/.j by A14,A15,PARTFUN1:def 6;
  reconsider j as Nat by A14;
A17: 1 <= j by A14,FINSEQ_3:25;
  reconsider j as Nat;
A18: x <> f/.i by A12,TARSKI:def 2;
  then
A19: j <> i by A14,A15,PARTFUN1:def 6;
A20: x <> f/.(i+1) by A12,TARSKI:def 2;
  then
A21: j <> i+1 by A14,A15,PARTFUN1:def 6;
  now
    per cases by A19,XXREAL_0:1;
    suppose
A22:  j+1 > len f;
A23:  j <= len f by A14,FINSEQ_3:25;
      len f <= j by A22,NAT_1:13;
      then
A24:  j = len f by A23,XXREAL_0:1;
      consider k be Nat such that
A25:  len f = 1 + k by A6,A8,NAT_1:10,XXREAL_0:2;
      reconsider k as Nat;
      1 < len f by A6,A8,XXREAL_0:2;
      then 1 <= k by A25,NAT_1:13;
      then
A26:  x in LSeg(f,k) by A16,A24,A25,TOPREAL1:21;
      i <= k by A2,A25,NAT_1:13;
      then i < k by A20,A16,A24,A25,XXREAL_0:1;
      then
A27:  i+1 <= k by NAT_1:13;
      now
        per cases by A27,XXREAL_0:1;
        suppose
A28:      i+1 = k;
          then i+(1+1) <= len f by A25;
          then LSeg(f,i) /\ LSeg(f,k) = {f/.(i+1)} by A1,A28,TOPREAL1:def 6;
          then x in {f/.(i+1)} by A13,A26,XBOOLE_0:def 4;
          hence contradiction by A20,TARSKI:def 1;
        end;
        suppose
          i+1 < k;
          then LSeg(f,i) misses LSeg(f,k) by TOPREAL1:def 7;
          hence contradiction by A13,A26,XBOOLE_0:3;
        end;
      end;
      hence contradiction;
    end;
    suppose that
A29:  i < j and
A30:  j+1 <= len f;
      i+1 <= j by A29,NAT_1:13;
      then i+1 < j by A21,XXREAL_0:1;
      then
A31:  LSeg(f,i) misses LSeg(f,j) by TOPREAL1:def 7;
      x in LSeg(f,j) by A16,A17,A30,TOPREAL1:21;
      hence contradiction by A13,A31,XBOOLE_0:3;
    end;
    suppose
A32:  j < i;
      then j+1 <= i+1 by XREAL_1:6;
      then j+1 <= len f by A6,XXREAL_0:2;
      then x in LSeg(f,j) by A16,A17,TOPREAL1:21;
      then
A33:  x in LSeg(f,i) /\ LSeg(f,j) by A13,XBOOLE_0:def 4;
A34:  j+1 <= i by A32,NAT_1:13;
      now
        per cases by A34,XXREAL_0:1;
        suppose
A35:      j+1 = i;
          then j+1+1 <= len f by A2,NAT_1:13;
          then j+(1+1) <= len f;
          then LSeg(f,i) /\ LSeg(f,j) = {f/.i} by A17,A35,TOPREAL1:def 6;
          hence contradiction by A18,A33,TARSKI:def 1;
        end;
        suppose
          j+1 < i;
          then LSeg(f,j) misses LSeg(f,i) by TOPREAL1:def 7;
          hence contradiction by A33,XBOOLE_0:4;
        end;
      end;
      hence contradiction;
    end;
  end;
  hence contradiction;
end;
