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 be FinSequence of TOP-REAL 2 for p be Point of TOP-REAL 2 st p
  in L~f holds R_Cut(f,p)/.1 = f/.1
proof
  let f be FinSequence of TOP-REAL 2;
  let p be Point of TOP-REAL 2;
  set i = Index(p,f);
  assume
A1: p in L~f;
  then
A2: 1 <= i by JORDAN3:8;
  i <= len f by A1,JORDAN3:8;
  then f <> {} by A2;
  then
A3: 1 in dom f by FINSEQ_5:6;
  p = f.1 or p <> f.1;
  then len R_Cut(f,p) = Index(p,f) or len R_Cut(f,p) = Index(p,f) + 1 by A1,
JORDAN3:25;
  then 1 <= len R_Cut(f,p) by A1,JORDAN3:8,NAT_1:11;
  hence (R_Cut(f,p))/.1 = R_Cut(f,p).1 by FINSEQ_4:15
    .= f.1 by A1,A2,JORDAN3:24
    .= f/.1 by A3,PARTFUN1:def 6;
end;
