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

theorem Th27:
  for f being circular unfolded s.c.c. FinSequence of TOP-REAL 2
  st len f > 4 holds LSeg(f,1) /\ L~(f/^1) = {f/.1,f/.2}
proof
  let f be circular unfolded s.c.c. FinSequence of TOP-REAL 2 such that
A1: len f > 4;
A2: 1+2 <= len f by A1,XXREAL_0:2;
  set h2 = f/^1;
A3: 1 <= len f by A1,XXREAL_0:2;
  then
A4: len h2 = len f - 1 by RFINSEQ:def 1;
  then
A5: len h2 + 1 = len f;
  len f > 3+1 by A1;
  then
A6: len h2 > 2+1 by A5,XREAL_1:6;
  then
A7: 1+1 <= len(f/^1) by XXREAL_0:2;
  set SFY = { LSeg(f/^1,i) : 1 < i & i+1 < len(f/^1) }, Reszta = union SFY;
A8: len(f/^1)-'1+1 <= len(f/^1) by A6,XREAL_1:235,XXREAL_0:2;
A9: 1 < len f by A1,XXREAL_0:2;
  for Z being set holds Z in {{}} iff ex X,Y being set st X in {LSeg(f,1)
  } & Y in SFY & Z = X /\ Y
  proof
    let Z be set;
    thus Z in {{}} implies ex X,Y being set st X in {LSeg(f,1)} & Y in SFY & Z
    = X /\ Y
    proof
      assume
A10:  Z in {{}};
      take X = LSeg(f,1), Y = LSeg(f,2+1);
      thus X in {LSeg(f,1)} by TARSKI:def 1;
      Y = LSeg(f/^1,2) by A3,SPPOL_2:4;
      hence Y in SFY by A6;
A11:  1+1 < 3;
      3+1 < len f by A1;
      then X misses Y by A11,GOBOARD5:def 4;
      then X /\ Y = {} by XBOOLE_0:def 7;
      hence Z = X /\ Y by A10,TARSKI:def 1;
    end;
    given X,Y being set such that
A12: X in {LSeg(f,1)} and
A13: Y in SFY and
A14: Z = X /\ Y;
A15: X = LSeg(f,1) by A12,TARSKI:def 1;
    consider i such that
A16: Y = LSeg(f/^1,i) and
A17: 1 < i and
A18: i+1 < len(f/^1) by A13;
A19: 1+1 < i+1 by A17,XREAL_1:6;
A20: i+1+1 < len f by A5,A18,XREAL_1:6;
    LSeg(f/^1,i) = LSeg(f,i+1) by A9,A17,SPPOL_2:4;
    then X misses Y by A15,A16,A20,A19,GOBOARD5:def 4;
    then Z = {} by A14,XBOOLE_0:def 7;
    hence thesis by TARSKI:def 1;
  end;
  then INTERSECTION({LSeg(f,1)},SFY) = {{}} by SETFAM_1:def 5;
  then
A21: LSeg(f,1) /\Reszta = union {{}} by SETFAM_1:25
    .= {} by ZFMISC_1:25;
A22: L~(f/^1) c= LSeg(f/^1,1) \/ LSeg(f/^1,len(f/^1)-'1) \/ Reszta
  proof
    let u be object;
    assume u in L~(f/^1);
    then consider e being set such that
A23: u in e and
A24: e in { LSeg(f/^1,i) : 1 <= i & i+1 <= len(f/^1) } by TARSKI:def 4;
    consider i such that
A25: e = LSeg(f/^1,i) and
A26: 1 <= i and
A27: i+1 <= len(f/^1) by A24;
    per cases by A26,A27,XXREAL_0:1;
    suppose
      i = 1;
      then u in LSeg(f/^1,1) \/ LSeg(f/^1,len(f/^1)-'1) by A23,A25,
XBOOLE_0:def 3;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
      i+1 = len(f/^1);
      then i = len(f/^1)-'1 by NAT_D:34;
      then u in LSeg(f/^1,1) \/ LSeg(f/^1,len(f/^1)-'1) by A23,A25,
XBOOLE_0:def 3;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
      1 < i & i+1 < len(f/^1);
      then e in SFY by A25;
      then u in Reszta by A23,TARSKI:def 4;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
A28: Reszta c= L~(f/^1)
  proof
    let u be object;
    assume u in Reszta;
    then consider e being set such that
A29: u in e and
A30: e in SFY by TARSKI:def 4;
    ex i st e = LSeg(f/^1,i) & 1 < i & i+1 < len(f/^1) by A30;
    then e in { LSeg(f/^1,i) : 1 <= i & i+1 <= len(f/^1) };
    hence thesis by A29,TARSKI:def 4;
  end;
  1+(len(f/^1)-'1) = len(f/^1) by A6,XREAL_1:235,XXREAL_0:2
    .= len f -' 1 by A1,A4,XREAL_1:233,XXREAL_0:2;
  then
A31: LSeg(f,1) /\ LSeg(f/^1,len(f/^1)-'1) = LSeg(f,1) /\ LSeg(f,len f-'1) by A3
,A7,NAT_D:55,SPPOL_2:4
    .= {f/.1} by A1,REVROT_1:30;
  1+1 <= len h2 by A6,NAT_1:13;
  then LSeg(f/^1,1) in { LSeg(f/^1,i) : 1 <= i & i+1 <= len(f/^1) };
  then
A32: LSeg(f/^1,1) c= L~(f/^1) by ZFMISC_1:74;
A33: LSeg(f,1) /\ LSeg(f/^1,1) = LSeg(f,1) /\ LSeg(f,1+1) by A3,SPPOL_2:4
    .= {f/.(1+1)} by A2,TOPREAL1:def 6;
  1 <= len(f/^1)-'1 by A7,NAT_D:55;
  then
  LSeg(f/^1,len(f/^1)-'1) in { LSeg(f/^1,i) : 1 <= i & i+1 <= len(f/^1) }
  by A8;
  then LSeg(f/^1,len(f/^1)-'1) c= L~(f/^1) by ZFMISC_1:74;
  then LSeg(f/^1,1) \/ LSeg(f/^1,len(f/^1)-'1) c= L~(f/^1) by A32,XBOOLE_1:8;
  then LSeg(f/^1,1) \/ LSeg(f/^1,len(f/^1)-'1) \/ Reszta c= L~(f/^1) by A28,
XBOOLE_1:8;
  then L~(f/^1) = LSeg(f/^1,1) \/ LSeg(f/^1,len(f/^1)-'1) \/ Reszta by A22,
XBOOLE_0:def 10;
  hence
  LSeg(f,1) /\ L~(f/^1) = LSeg(f,1) /\ (LSeg(f/^1,1) \/ LSeg(f/^1,len(f/^
  1)-'1)) \/ {} by A21,XBOOLE_1:23
    .= {f/.1} \/ {f/.2} by A33,A31,XBOOLE_1:23
    .= {f/.1,f/.2} by ENUMSET1:1;
end;
