reserve i,j,k,n,m for Nat;
reserve p,q for Point of TOP-REAL 2;
reserve G for Go-board;
reserve C for Subset of TOP-REAL 2;

theorem
  for g being FinSequence of TOP-REAL 2, p being Point of TOP-REAL 2 st
g/.1 <> p & ((g/.1)`1 = p`1 or (g/.1)`2 = p`2) & g is being_S-Seq & LSeg(p,g/.1
  ) /\ L~g={ g/.1 } holds <*p*>^g is being_S-Seq
proof
  let g be FinSequence of TOP-REAL 2, p be Point of TOP-REAL 2 such that
A1: g/.1 <> p and
A2: (g/.1)`1 = p`1 or (g/.1)`2 = p`2 and
A3: g is being_S-Seq and
A4: LSeg(p,g/.1) /\ L~g={ g/.1 };
  set f = <*p,g/.1*>;
A5: f is being_S-Seq by A1,A2,SPPOL_2:43;
  reconsider g9 = g as S-Sequence_in_R2 by A3;
A6: 1 in dom g9 by FINSEQ_5:6;
A7: len f = 1+1 by FINSEQ_1:44;
  then AB: len f -' 1 = 1 by NAT_D:34;
AA:  1 in dom f by FINSEQ_3:25,A7;
  then
A8: mid(f,1,len f-'1) = <*f.1*> by AB,FINSEQ_6:193
    .= <*f/.1*> by AA,PARTFUN1:def 6
    .= <*p*> by FINSEQ_4:17;
A9: L~f /\ L~g ={ g/.1 } by A4,SPPOL_2:21
    .={g.1} by A6,PARTFUN1:def 6;
  f.len f = g/.1 by A7
    .= g.1 by A6,PARTFUN1:def 6;
  hence thesis by A3,A5,A9,A8,JORDAN3:45;
end;
