
theorem
  for f being FinSequence of TOP-REAL 2,
  p, q being Point of TOP-REAL 2 st p in L~f & q in L~f & f is being_S-Seq
  holds L~B_Cut(f,p,q) c= L~f
proof
  let f be FinSequence of TOP-REAL 2, p, q be Point of TOP-REAL 2 such that
A1: p in L~f and
A2: q in L~f and
A3: f is being_S-Seq;
A4: f is one-to-one by A3;
  per cases;
  suppose p = q;
    then B_Cut(f,p,q) = <*p*> by A1,A4,Th21;
    then len B_Cut(f,p,q) = 1 by FINSEQ_1:39;
    then L~B_Cut(f,p,q) = {} by TOPREAL1:22;
    hence thesis;
  end;
  suppose p <> q & (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));
    hence thesis by A1,A2,Lm3;
  end;
  suppose that
A5: p <> q and
A6: not(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));
A7: B_Cut(f,p,q)=Rev R_Cut(L_Cut(f,q),p) by A6,JORDAN3:def 7;
A8: Index(q,f) < Index(p,f)
    or Index(p,f)=Index(q,f) & not LE p,q,f/.Index(p,f),f/.(Index(p,f)+1)
    by A6,XXREAL_0:1;
A9: now
      assume that
A10:  Index(p,f)=Index(q,f) and
A11:  not LE p,q,f/.Index(p,f),f/.(Index(p,f)+1);
A12:  1 <= Index(p,f) by A1,JORDAN3:8;
A13:  Index(p,f) < len f by A1,JORDAN3:8;
      then
A14:  Index(p,f)+1 <= len f by NAT_1:13;
      then
A15:  LSeg(f,Index(p,f)) = LSeg(f/.Index(p,f),f/.(Index(p,f)+1))
      by A12,TOPREAL1:def 3;
      then
A16:  p in LSeg(f/.Index(p,f),f/.(Index(p,f)+1)) by A1,JORDAN3:9;
A17:  q in LSeg(f/.Index(p,f),f/.(Index(p,f)+1)) by A2,A10,A15,JORDAN3:9;
A18:  Index(p,f) in dom f by A12,A13,FINSEQ_3:25;
      1 <= Index(p,f)+1 by NAT_1:11;
      then
A19:  Index(p,f)+1 in dom f by A14,FINSEQ_3:25;
      Index(p,f)+0 <> Index(p,f)+1;
      then f/.Index(p,f)<>f/.(Index(p,f)+1) by A4,A18,A19,PARTFUN2:10;
      then LT q,p,f/.Index(p,f), f/.(Index(p,f)+1) by A11,A16,A17,JORDAN3:28;
      hence LE q,p,f/.Index(q,f),f/.(Index(q,f)+1) by A10;
    end;
    then
A20: Rev B_Cut(f,q,p) = B_Cut(f,p,q) by A1,A2,A7,A8,JORDAN3:def 7;
    L~B_Cut(f,q,p) c= L~f by A1,A2,A5,A8,A9,Lm3;
    hence thesis by A20,SPPOL_2:22;
  end;
end;
