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 Th35:
  for f being circular s.c.c. FinSequence of TOP-REAL 2 st len f >
4 for i,j being Nat st 1 <= i & i < j & j < len f holds f/.i <> f/.j
proof
  let f be circular s.c.c. FinSequence of TOP-REAL 2 such that
A1: len f > 4;
  let i,j be Nat such that
A2: 1 <= i and
A3: i < j and
A4: j < len f and
A5: f/.i = f/.j;
A6: j+1 <= len f by A4,NAT_1:13;
A7: i+1 <= j & i <> 0 by A2,A3,NAT_1:13;
  1 <= j by A2,A3,XXREAL_0:2;
  then
A8: f/.j in LSeg(f,j) by A6,TOPREAL1:21;
A9: i < len f by A3,A4,XXREAL_0:2;
  then i+1 <= len f by NAT_1:13;
  then
A10: f/.i in LSeg(f,i) by A2,TOPREAL1:21;
  i <= 2 implies i = 0 or ... or i = 2;
  then per cases by A7,XXREAL_0:1;
  suppose that
A11: i+1 = j and
A12: i = 1;
A13: len f -' 1 + 1 = len f by A1,XREAL_1:235,XXREAL_0:2;
    j+1+1 < len f by A1,A11,A12;
    then
A14: j+1 < len f -' 1 by A13,XREAL_1:6;
    len f -' 1 < len f by A13,XREAL_1:29;
    then LSeg(f,j) misses LSeg(f,len f -' 1) by A11,A12,A14,GOBOARD5:def 4;
    then
A15: LSeg(f,j) /\ LSeg(f,len f -' 1) = {} by XBOOLE_0:def 7;
A16: f/.i = f/.len f by A12,FINSEQ_6:def 1;
    1+1 <= len f by A1,XXREAL_0:2;
    then 1 <= len f -' 1 by A13,XREAL_1:6;
    then f/.i in LSeg(f,len f -' 1) by A13,A16,TOPREAL1:21;
    hence contradiction by A5,A8,A15,XBOOLE_0:def 4;
  end;
  suppose that
A17: i+1 = j and
A18: i = 1+1;
A19: i -' 1 + 1 = i by A2,XREAL_1:235;
    j+1 < len f by A1,A17,A18;
    then LSeg(f,i -' 1) misses LSeg(f,j) by A3,A19,GOBOARD5:def 4;
    then
A20: LSeg(f,i -' 1) /\ LSeg(f,j) = {} by XBOOLE_0:def 7;
    f/.i in LSeg(f,i -' 1) by A9,A18,A19,TOPREAL1:21;
    hence contradiction by A5,A8,A20,XBOOLE_0:def 4;
  end;
  suppose that
A21: i > 1+1;
A22: i -' 1 + 1 = i by A2,XREAL_1:235;
    then
A23: 1 < i -' 1 by A21,XREAL_1:6;
    then LSeg(f,i-'1) misses LSeg(f,j) by A3,A4,A22,GOBOARD5:def 4;
    then
A24: LSeg(f,i-'1) /\ LSeg(f,j) = {} by XBOOLE_0:def 7;
    f/.i in LSeg(f,i-' 1) by A9,A22,A23,TOPREAL1:21;
    hence contradiction by A5,A8,A24,XBOOLE_0:def 4;
  end;
  suppose that
A25: i+1 < j and
A26: i <> 1;
    1 < i by A2,A26,XXREAL_0:1;
    then LSeg(f,i) misses LSeg(f,j) by A4,A25,GOBOARD5:def 4;
    then LSeg(f,i) /\ LSeg(f,j) = {} by XBOOLE_0:def 7;
    hence contradiction by A5,A8,A10,XBOOLE_0:def 4;
  end;
  suppose that
A27: i+1 < j and
A28: j+1 <> len f;
    j+1 < len f by A6,A28,XXREAL_0:1;
    then LSeg(f,i) misses LSeg(f,j) by A27,GOBOARD5:def 4;
    then LSeg(f,i) /\ LSeg(f,j) = {} by XBOOLE_0:def 7;
    hence contradiction by A5,A8,A10,XBOOLE_0:def 4;
  end;
  suppose that
A29: i+1 < j and
A30: i = 1 and
A31: j+1 = len f;
A32: j < len f by A31,NAT_1:13;
A33: j-'1+1 = j by A2,A3,XREAL_1:235,XXREAL_0:2;
    then
A34: i+1 <= j-'1 by A29,NAT_1:13;
    i+1 <> j-'1 by A1,A30,A31,A33;
    then i+1 < j-'1 by A34,XXREAL_0:1;
    then LSeg(f,1) misses LSeg(f,j-'1) by A30,A33,A32,GOBOARD5:def 4;
    then
A35: LSeg(f,1) /\ LSeg(f,j-'1) = {} by XBOOLE_0:def 7;
    1 <= j-'1 by A30,A34,XXREAL_0:2;
    then f/.j in LSeg(f,j-'1) by A4,A33,TOPREAL1:21;
    hence contradiction by A5,A10,A30,A35,XBOOLE_0:def 4;
  end;
end;
