reserve n for Nat;

theorem Th3:
  for f be FinSequence of TOP-REAL 2 for p,q be Point of TOP-REAL 2
  holds B_Cut (f,p,q) <> {}
proof
  let f be FinSequence of TOP-REAL 2;
  let p,q be Point of TOP-REAL 2;
A1: R_Cut(L_Cut(f,q),p) <> {} by JORDAN1J:44;
  per cases;
  suppose
    p in L~f & q in L~f & Index(p,f)<Index(q,f) or Index(p,f)=Index(q,
    f) & LE p,q,f/.(Index(p,f)),f/.(Index(p,f)+1);
    then B_Cut(f,p,q) = R_Cut(L_Cut(f,p),q) by JORDAN3:def 7;
    hence thesis by JORDAN1J:44;
  end;
  suppose
    not (p in L~f & q in L~f & Index(p,f)<Index(q,f) or Index(p,f)=
    Index(q,f) & LE p,q,f/.(Index(p,f)),f/.(Index(p,f)+1));
    then B_Cut(f,p,q) = Rev R_Cut(L_Cut(f,q),p) by JORDAN3:def 7;
    hence thesis by A1;
  end;
end;
