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 L_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
A4: len g in dom g by FINSEQ_3:25;
  assume p in L~g;
  then Index(p,g) < len g by JORDAN3:8;
  then
A5: Index(p,g)+1 <= len g by NAT_1:13;
  1<=Index(p,g)+1 by NAT_1:11;
  then
A6: Index(p,g)+1 in dom g by A5,FINSEQ_3:25;
  per cases;
  suppose
    p<>g.(Index(p,g)+1);
    then
A7: L_Cut(g,p) = <*p*>^mid(g,Index(p,g)+1,len g) by JORDAN3:def 3;
    mid(g,Index(p,g)+1,len g) is_in_the_area_of f by A1,A4,A6,SPRECT_2:22;
    hence thesis by A2,A7,SPRECT_2:24;
  end;
  suppose
    p=g.(Index(p,g)+1);
    then L_Cut(g,p) = mid(g,Index(p,g)+1,len g) by JORDAN3:def 3;
    hence thesis by A1,A4,A6,SPRECT_2:22;
  end;
end;
