
theorem Th21:
  for f being FinSequence of TOP-REAL 2,
  p being Point of TOP-REAL 2 st p in L~f & f is one-to-one
  holds B_Cut(f,p,p) = <*p*>
proof
  let f be FinSequence of TOP-REAL 2, p be Point of TOP-REAL 2;
  assume that
A1: p in L~f and
A2: f is one-to-one;
A3: Index(p,f) <> Index(p,f)+1;
A4: Index(p,f) < len f by A1,JORDAN3:8;
A5: 1 <= Index(p,f) by A1,JORDAN3:8;
A6: Index(p,f) + 1 <= len f by A4,NAT_1:13;
  then
A7: Index(p,f) in dom f by A5,SEQ_4:134;
A8: Index(p,f) + 1 in dom f by A5,A6,SEQ_4:134;
  p in LSeg(f,Index(p,f)) by A1,JORDAN3:9;
  then p in LSeg(f/.Index(p,f), f/.(Index(p,f)+1)) by A5,A6,TOPREAL1:def 3;
  then
A9: LE p,p,f/.Index(p,f),f/.(Index(p,f)+1) by A2,A3,A7,A8,Th9,PARTFUN2:10;
  (L_Cut(f,p)).1 = p by A1,JORDAN3:23;
  then R_Cut(L_Cut(f,p),p) = <*p*> by JORDAN3:def 4;
  hence thesis by A9,JORDAN3:def 7;
end;
