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, p being Point of TOP-REAL 2
st g is_in_the_area_of f & <*p*> is_in_the_area_of f & g is being_S-Seq & p in
  L~g holds R_Cut(g,p) is_in_the_area_of f
proof
  let f, g be FinSequence of TOP-REAL 2, p be Point of TOP-REAL 2 such that
A1: g is_in_the_area_of f and
A2: <*p*> is_in_the_area_of f and
A3: g is being_S-Seq;
  2 <= len g by A3,TOPREAL1:def 8;
  then 1 <= len g by XXREAL_0:2;
  then
A5: 1 in dom g by FINSEQ_3:25;
  assume
A6: p in L~g;
  then
A7: Index(p,g) < len g by JORDAN3:8;
  1<=Index(p,g) by A6,JORDAN3:8;
  then
A8: Index(p,g) in dom g by A7,FINSEQ_3:25;
  per cases;
  suppose
    p<>g.1;
    then
A9: R_Cut(g,p) = mid(g,1,Index(p,g))^<*p*> by JORDAN3:def 4;
    mid(g,1,Index(p,g)) is_in_the_area_of f by A1,A5,A8,SPRECT_2:22;
    hence thesis by A2,A9,SPRECT_2:24;
  end;
  suppose
    p=g.1;
    then R_Cut(g,p) = <*g.1*> by JORDAN3:def 4
      .= mid(g,1,1) by A5,FINSEQ_6:193;
    hence thesis by A1,A5,SPRECT_2:22;
  end;
end;
