reserve r1,r2 for Real;
reserve n,i,i1,i2,j for Nat;
reserve D for non empty set;
reserve f for FinSequence of D;

theorem
  for f be FinSequence of TOP-REAL 2 for p be Point of TOP-REAL 2 st p
  in L~f holds ( p<>f.1 implies len R_Cut(f,p)=Index(p,f)+1 ) & ( p=f.1 implies
  len R_Cut(f,p)=Index(p,f) )
proof
  let f be FinSequence of TOP-REAL 2;
  let p be Point of TOP-REAL 2;
  assume
A1: p in L~f;
  then consider i be Nat such that
A2: 1 <= i and
A3: i+1 <= len f and
  p in LSeg(f,i) by SPPOL_2:13;
A4: 1<=Index(p,f) by A1,Th8;
A5: Index(p,f)<=len f by A1,Th8;
  i<=len f by A3,NAT_D:46;
  then
A6: 1<=len f by A2,XXREAL_0:2;
  now
    per cases;
    case
      p<>f.1;
      then R_Cut(f,p)=mid(f,1,Index(p,f))^<*p*> by Def4;
      hence len R_Cut(f,p) = len mid(f,1,Index(p,f))+len <*p*> by FINSEQ_1:22
        .= len mid(f,1,Index(p,f))+1 by FINSEQ_1:39
        .= Index(p,f)-'1+1+1 by A6,A4,A5,FINSEQ_6:118
        .= Index(p,f)+1 by A1,Th8,XREAL_1:235;
    end;
    case
A7:   p=f.1;
      len f > i by A3,NAT_1:13;
      then len f > 1 by A2,XXREAL_0:2;
      then
A8:   len f >= 1+1 by NAT_1:13;
      1 in dom f by A3,CARD_1:27,FINSEQ_5:6;
      then
A9:   p = f/.1 by A7,PARTFUN1:def 6;
      R_Cut(f,p)=<*p*> by A7,Def4;
      hence len R_Cut(f,p)= 1 by FINSEQ_1:39
        .= Index(p,f) by A8,A9,Th11;
    end;
  end;
  hence thesis;
end;
