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 f, g being FinSequence of TOP-REAL 2 st f is being_S-Seq & g is
being_S-Seq & ((f/.len f)`1 = (g/.1)`1 or (f/.len f)`2 = (g/.1)`2) & L~f misses
L~g & LSeg(f/.len f,g/.1) /\ L~f={ f/.len f } & LSeg(f/.len f,g/.1) /\ L~g={ g
  /.1 } holds f^g is being_S-Seq
proof
  let f,g be FinSequence of TOP-REAL 2 such that
A1: f is being_S-Seq and
A2: g is being_S-Seq and
A3: (f/.len f)`1 = (g/.1)`1 or (f/.len f)`2 = (g/.1)`2 and
A4: L~f misses L~g and
A5: LSeg(f/.len f,g/.1) /\ L~f={ f/.len f } and
A6: LSeg(f/.len f,g/.1) /\ L~g={ g/.1 };
A7: len g >= 2 by A2,TOPREAL1:def 8;
  then
A8: len g >= 1 by XXREAL_0:2;
  then 1 in dom g by FINSEQ_3:25;
  then
A9: g/.1 in L~g by A7,GOBOARD1:1;
  set h = <*f/.len f*>^g;
A10: len f >= 2 by A1,TOPREAL1:def 8;
  then 1 <= len f by XXREAL_0:2;
  then
A11: len f in dom f by FINSEQ_3:25;
  then
A12: f.len f = f/.len f by PARTFUN1:def 6
    .=h.1 by FINSEQ_1:41;
A13: len h = len<*f/.len f*>+len g by FINSEQ_1:22
    .= len g + 1 by FINSEQ_1:39;
  then len h >= 1+1 by A8,XREAL_1:6;
  then
A14: mid(h,2,len h) = (h/^(1+1-'1))|(len h-'2+1) by FINSEQ_6:def 3
    .= (h/^1)|(len h-'2+1) by NAT_D:34
    .= g|(len h-'2+1) by FINSEQ_6:45
    .= g|(len h-'1-'1+1) by NAT_D:45
    .= g|(len g-'1+1) by A13,NAT_D:34
    .= g|len g by A7,XREAL_1:235,XXREAL_0:2
    .= g by FINSEQ_1:58;
  f/.len f in L~f by A10,A11,GOBOARD1:1;
  then f/.len f <> g/.1 by A4,A9,XBOOLE_0:3;
  then
A15: h is being_S-Seq by A2,A3,A6,SPRECT_2:60;
  L~f /\ L~h = L~f /\ (LSeg(f/.len f,g/.1) \/ L~g) by A2,SPPOL_2:20
    .= L~f /\ LSeg(f/.len f,g/.1) \/ L~f /\ L~g by XBOOLE_1:23
    .= L~f /\ LSeg(f/.len f,g/.1) \/ {} by A4
    .={h.1} by A5,A11,A12,PARTFUN1:def 6;
  hence thesis by A1,A12,A15,A14,JORDAN3:38;
end;
