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 Th23:
  for f being FinSequence of TOP-REAL 2, p being Point of TOP-REAL
2 st p in L~f holds (L_Cut(f,p)).1=p & for i st 1<i & i<=len L_Cut(f,p) holds (
p = f.(Index(p,f)+1) implies (L_Cut(f,p)).i=f.(Index(p,f)+i)) & (p <> f.(Index(
  p,f)+1) implies (L_Cut(f,p)).i=f.(Index(p,f)+i-1))
proof
  let f be FinSequence of TOP-REAL 2, p be Point of TOP-REAL 2;
  assume
A1: p in L~f;
  then Index(p,f) < len f by Th8;
  then
A2: Index(p,f)+1<=len f by NAT_1:13;
A3: f is non empty by A1,CARD_1:27,TOPREAL1:22;
  now
    per cases;
    suppose
A4:   p = f.(Index(p,f)+1);
      1 in dom f by A3,FINSEQ_5:6;
      then
A5:   1 <= len f by FINSEQ_3:25;
      Index(p,f) < len f by A1,Th8;
      then
A6:   Index(p,f)+1 <= len f by NAT_1:13;
A7:   1 <= Index(p,f)+1 by NAT_1:11;
      L_Cut(f,p)=mid(f,Index(p,f)+1,len f) by A4,Def3;
      hence (L_Cut(f,p)).1 = p by A4,A7,A6,A5,FINSEQ_6:118;
    end;
    suppose
      p<> f.(Index(p,f)+1);
      then L_Cut(f,p)=<*p*>^mid(f,Index(p,f)+1,len f) by Def3;
      hence (L_Cut(f,p)).1=p by FINSEQ_1:41;
    end;
  end;
  hence (L_Cut(f,p)).1=p;
  let i;
  assume that
A8: 1<i and
A9: i<=len L_Cut(f,p);
A10: len <*p*><=i by A8,FINSEQ_1:40;
A11: 1<=Index(p,f)+1 by NAT_1:11;
  then
A12: 1<=len f by A2,XXREAL_0:2;
  then len mid(f,Index(p,f)+1,len f)=len f -'(Index(p,f)+1)+1 by A11,A2,
FINSEQ_6:118;
  then
A13: len <*p*>+len mid(f,Index(p,f)+1,len f) =1+(len f -'(Index(p,f)+1)+1)
  by FINSEQ_1:40
    .=1+(len f -(Index(p,f)+1)+1) by A2,XREAL_1:233
    .=(len f -Index(p,f))+1;
A14: (i-'1)+1=i-1+1 by A8,XREAL_1:233
    .=i;
A15: 1<=i-1 by A8,SPPOL_1:1;
  then
A16: 1<=i-'1 by NAT_D:39;
  hereby
    assume p = f.(Index(p,f)+1);
    then L_Cut(f,p)=mid(f,Index(p,f)+1,len f) by Def3;
    hence (L_Cut(f,p)).i = f.(i+(Index(p,f)+1)-'1) by A8,A9,A11,A2,A12,
FINSEQ_6:118
      .= f.(i+Index(p,f)+1-'1)
      .= f.(Index(p,f)+i) by NAT_D:34;
  end;
A17: i <=i+Index(p,f) by NAT_1:11;
  assume p <> f.(Index(p,f)+1);
  then
A18: L_Cut(f,p)=<*p*>^mid(f,Index(p,f)+1,len f) by Def3;
  then i<=len <*p*>+len mid(f,Index(p,f)+1,len f) by A9,FINSEQ_1:22;
  then i-1<=len f -Index(p,f)+1-1 by A13,XREAL_1:9;
  then
A19: i-'1<=len f -(Index(p,f)+1)+1 by A15,NAT_D:39;
  len <*p*><i by A8,FINSEQ_1:39;
  then (L_Cut(f,p)).i =mid(f,Index(p,f)+1,len f).(i-len <*p*>) by A9,A18,
FINSEQ_6:108
    .=mid(f,Index(p,f)+1,len f).(i-'len <*p*>) by A10,XREAL_1:233
    .=mid(f,Index(p,f)+1,len f).(i-'1) by FINSEQ_1:39
    .=f.((i-'1)+(Index(p,f)+1)-'1) by A11,A2,A16,A19,FINSEQ_6:122
    .=f.(Index(p,f)+i-'1) by A14;
  hence thesis by A8,A17,XREAL_1:233,XXREAL_0:2;
end;
