reserve i,j,k,m,n for Nat,
  D for non empty set,
  p for Element of D,
  f for FinSequence of D;
reserve D for non empty set,
  p for Element of D,
  f for FinSequence of D;
reserve f for circular FinSequence of D;
reserve f,g for FinSequence of TOP-REAL 2;
reserve p for Point of TOP-REAL 2,
  f for FinSequence of TOP-REAL 2;
reserve f for circular FinSequence of TOP-REAL 2;

theorem Th30:
  for f being circular s.c.c. FinSequence of TOP-REAL 2 st len f >
  4 holds LSeg(f,len f -' 1) /\ LSeg(f,1) = {f/.1}
proof
  let f be circular s.c.c. FinSequence of TOP-REAL 2;
  assume
A1: len f > 4;
  then
A2: len f -' 1 + 1 = len f by XREAL_1:235,XXREAL_0:2;
A3: len f >= 1+1 by A1,XXREAL_0:2;
  then
A4: 1 <= len f -' 1 by NAT_D:55;
A5: len f >= 1+1+1 by A1,XXREAL_0:2;
  thus LSeg(f,len f -' 1) /\ LSeg(f,1) c= {f/.1}
  proof
    assume not LSeg(f,len f -' 1) /\ LSeg(f,1) c= {f/.1};
    then consider p being Point of TOP-REAL 2 such that
A6: p in LSeg(f,len f -' 1) /\ LSeg(f,1) and
A7: not p in {f/.1};
A8: p <> f/.1 by A7,TARSKI:def 1;
A9: LSeg(f,len f -' 1) = LSeg(f/.(len f -' 1),f/.len f) & LSeg(f,1) = LSeg
    (f/.1, f/.(1+1)) by A3,A2,A4,TOPREAL1:def 3;
A10: f/.len f = f/.1 by FINSEQ_6:def 1;
    per cases by A6,A9,A8,A10,JGRAPH_1:16;
    suppose
A11:  f/.(1+1) in LSeg(f,len f -' 1);
A12:  f/.(1+1) in LSeg(f,1+1) by A5,TOPREAL1:21;
      3+1 = 4;
      then 1+1+1 < len f - 1 by A1,XREAL_1:20;
      then
A13:  1+1+1 < len f -' 1 by A1,XREAL_1:233,XXREAL_0:2;
      len f -' 1 < len f by A4,NAT_D:51;
      then LSeg(f,1+1) misses LSeg(f,len f -' 1) by A13,GOBOARD5:def 4;
      hence contradiction by A11,A12,XBOOLE_0:3;
    end;
    suppose
A14:  f/.(len f -' 1) in LSeg(f,1);
A15:  len f -' 2+1 = len f -' 1 -' 1+1 by NAT_D:45
        .= len f -' 1 by A3,NAT_D:55,XREAL_1:235;
      then
A16:  len f -' 2+1 < len f by A4,NAT_D:51;
      2 + 2 < len f by A1;
      then 1+1 < len f - 2 by XREAL_1:20;
      then 1+1 < len f -' 2 by A1,XREAL_1:233,XXREAL_0:2;
      then
A17:  LSeg(f,1) misses LSeg(f,len f -' 2) by A16,GOBOARD5:def 4;
      1 <= len f - 2 by A5,XREAL_1:19;
      then 1 <= len f -' 2 by NAT_D:39;
      then f/.(len f -' 1) in LSeg(f,len f -' 2) by A15,A16,TOPREAL1:21;
      hence contradiction by A14,A17,XBOOLE_0:3;
    end;
  end;
  let x be object;
  assume x in {f/.1};
  then
A18: x = f/.1 by TARSKI:def 1;
  then x = f/.len f by FINSEQ_6:def 1;
  then
A19: x in LSeg(f,len f -' 1) by A2,A4,TOPREAL1:21;
  x in LSeg(f,1) by A3,A18,TOPREAL1:21;
  hence thesis by A19,XBOOLE_0:def 4;
end;
