reserve E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  i, j, m, n for Nat,
  p for Point of TOP-REAL 2;

theorem Th3:
  for f be FinSequence of TOP-REAL 2 for p be Point of TOP-REAL 2
  st 1 <= len f & 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;
  assume that
A1: 1 <= len f and
A2: p in L~f;
A3: 1 <= Index(p,f) by A2,JORDAN3:8;
A4: 1 in dom f by A1,FINSEQ_3:25;
  now
    per cases;
    suppose
      p<>f.1;
      then
A5:   R_Cut(f,p) = mid(f,1,Index(p,f))^<*p*> by JORDAN3:def 4;
A6:   Index(p,f) < len f by A2,JORDAN3:8;
      then Index(p,f) in dom f by A3,FINSEQ_3:25;
      then len mid(f,1,Index(p,f)) >= 1 by A4,SPRECT_2:5;
      hence R_Cut(f,p).1 = mid(f,1,Index(p,f)).1 by A5,FINSEQ_6:109
        .= f.1 by A1,A3,A6,FINSEQ_6:118;
    end;
    suppose
A7:   p = f.1;
      then R_Cut(f,p) = <*p*> by JORDAN3:def 4;
      hence thesis by A7;
    end;
  end;
  hence thesis;
end;
