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 Th29:
  for f be FinSequence of TOP-REAL 2 for p,q be Point of TOP-REAL
  2 st p in L~f & q in L~f & Index(p,f)<Index(q,f) holds q in L~L_Cut(f,p)
proof
  let f be FinSequence of TOP-REAL 2;
  let p,q be Point of TOP-REAL 2;
  assume that
A1: p in L~f and
A2: q in L~f and
A3: Index(p,f)<Index(q,f);
A4: Index(q,f)<len f by A2,Th8;
  then
A5: Index(q,f)-Index(p,f)<=len f -Index(p,f) by XREAL_1:9;
  then
A6: Index(q,f)-Index(p,f)+1<=len f -Index(p,f)+1 by XREAL_1:6;
  Index(q,f)-Index(p,f)<=len f -Index(p,f)-1+1 by A4,XREAL_1:9;
  then
A7: Index(q,f)-'Index(p,f)<=len f - (Index(p,f)+1)+1 by A3,XREAL_1:233;
  set i1 = Index(q,f)-'Index(p,f)+1;
A8: 1<=Index(p,f)+1 by NAT_1:11;
A9: Index(p,f)+1<=Index(q,f) by A3,NAT_1:13;
  then
A10: Index(p,f)+1-Index(p,f)<=Index(q,f)-Index(p,f) by XREAL_1:9;
  then
A11: 1<=Index(q,f)-'Index(p,f) by XREAL_0:def 2;
  then
A12: 1<=Index(q,f)-'Index(p,f)+1 by NAT_D:48;
  1+1<=Index(q,f)-'Index(p,f)+1 by A11,XREAL_1:6;
  then
A13: 1<Index(q,f)-'Index(p,f)+1 by XXREAL_0:2;
  then
A14: len <*p*><Index(q,f)-'Index(p,f)+1 by FINSEQ_1:40;
  then
A15: len <*p*><Index(q,f)-'Index(p,f)+1+1 by NAT_1:13;
A16: Index(p,f)+1<=len f by A4,A9,XXREAL_0:2;
A17: 1<=Index(q,f) by A2,Th8;
  then 1<len f by A4,XXREAL_0:2;
  then
  len mid(f,Index(p,f)+1,len f) = len f -'(Index(p,f)+1)+1
   by A8,A16,FINSEQ_6:118;
  then
A18: 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 A4,A9,XREAL_1:233,XXREAL_0:2
    .= (len f -Index(p,f))+1;
  then
A19: Index(q,f)-'Index(p,f)+1<=len <*p*>+len mid(f,Index(p,f)+1,len f) by A3,A6
,XREAL_1:233;
  per cases;
  suppose
A20: p = f.(Index(p,f)+1);
    then
A21: len L_Cut(f,p)=len f -Index(p,f) by A1,Th26;
    then len L_Cut(f,p)>=Index(q,f)-'Index(p,f) by A3,A5,XREAL_1:233;
    then (L_Cut(f,p))/.(Index(q,f)-'Index(p,f)) = L_Cut(f,p).(Index(q,f)-'
    Index(p,f)) by A11,FINSEQ_4:15
      .= mid(f,Index(p,f)+1,len f).(Index(q,f)-'Index(p,f)) by A20,Def3
      .= f.(Index(p,f)+1+(Index(q,f)-'Index(p,f))-1)
       by A11,A8,A16,A7,FINSEQ_6:122
      .= f.(Index(p,f)+1+(Index(q,f)-Index(p,f))-1) by A3,XREAL_1:233
      .= f.Index(q,f);
    then
A22: (L_Cut(f,p))/.(Index(q,f)-'Index(p,f))=f/.Index(q,f) by A2,A4,Th8,
FINSEQ_4:15;
    1<=Index(q,f) by A2,Th8;
    then
A23: 1<=Index(q,f)+1 by NAT_D:48;
A24: q in LSeg(f,Index(q,f)) by A2,Th9;
A25: Index(q,f)<len f by A2,Th8;
    then
A26: Index(q,f)+1<=len f by NAT_1:13;
    then
A27: Index(q,f)+1-Index(p,f)<=len f -Index(p,f) by XREAL_1:9;
    then Index(q,f)-Index(p,f)+1<=len f -Index(p,f);
    then
A28: Index(q,f)-'Index(p,f)+1<=len f - (Index(p,f)+1)+1 by A3,XREAL_1:233;
    Index(q,f)+1-Index(p,f)<=len f -Index(p,f) by A26,XREAL_1:9;
    then Index(q,f)-Index(p,f)+1<=len f -Index(p,f);
    then
A29: i1<=len L_Cut(f,p) by A10,A21,XREAL_0:def 2;
A30: Index(q,f)+1<=len f by A25,NAT_1:13;
    Index(q,f)-Index(p,f)+1<=len f -Index(p,f) by A27;
    then len L_Cut(f,p)>=Index(q,f)-'Index(p,f)+1 by A3,A21,XREAL_1:233;
    then (L_Cut(f,p))/.(Index(q,f)-'Index(p,f)+1) = L_Cut(f,p).(Index(q,f)-'
    Index(p,f)+1) by A13,FINSEQ_4:15
      .= (mid(f,Index(p,f)+1,len f)).(Index(q,f)-'Index(p,f)+1) by A20,Def3
      .= f.(Index(p,f)+1+(Index(q,f)-'Index(p,f)+1)-1) by A12,A8,A16,A28,
FINSEQ_6:122
      .= f.(Index(p,f)+1+(Index(q,f)-Index(p,f)+1)-1) by A3,XREAL_1:233
      .= f/.(Index(q,f)+1) by A23,A30,FINSEQ_4:15;
    then q in LSeg((L_Cut(f,p))/.((Index(q,f)-'Index(p,f))), (L_Cut(f,p))/.((
    Index(q,f)-'Index(p,f)+1))) by A17,A22,A26,A24,TOPREAL1:def 3;
    hence thesis by A11,A29,SPPOL_2:15;
  end;
  suppose that
A31: p <> f.(Index(p,f)+1);
A32: len L_Cut(f,p)=len f -Index(p,f)+1 by A1,A31,Th26;
    then len L_Cut(f,p)>=Index(q,f)-'Index(p,f)+1 by A3,A6,XREAL_1:233;
    then (L_Cut(f,p))/.(Index(q,f)-'Index(p,f)+1) = L_Cut(f,p).(Index(q,f)-'
    Index(p,f)+1) by FINSEQ_4:15,NAT_1:11
      .= (<*p*>^mid(f,Index(p,f)+1,len f)).(Index(q,f)-'Index(p,f)+1) by A31
,Def3
      .= mid(f,Index(p,f)+1,len f).(Index(q,f)-'Index(p,f)+1-len <*p*>) by A14
,A19,FINSEQ_6:108
      .= mid(f,Index(p,f)+1,len f).(Index(q,f)-'Index(p,f)+1-1) by FINSEQ_1:40
      .= f.(Index(p,f)+1+(Index(q,f)-'Index(p,f))-1)
       by A11,A8,A16,A7,FINSEQ_6:122
      .= f.(Index(p,f)+1+(Index(q,f)-Index(p,f))-1) by A3,XREAL_1:233
      .= f.Index(q,f);
    then
A33: (L_Cut(f,p))/.(Index(q,f)-'Index(p,f)+1)=f/.Index(q,f) by A2,A4,Th8,
FINSEQ_4:15;
A34: Index(q,f)<len f by A2,Th8;
    then
A35: Index(q,f)+1<=len f by NAT_1:13;
    then
A36: Index(q,f)+1-Index(p,f)<=len f -Index(p,f) by XREAL_1:9;
    then Index(q,f)-Index(p,f)+1<=len f -Index(p,f);
    then
A37: Index(q,f)-'Index(p,f)+1<=len f - (Index(p,f)+1)+1 by A3,XREAL_1:233;
    Index(q,f)+1-Index(p,f)<=len f -Index(p,f) by A35,XREAL_1:9;
    then Index(q,f)-Index(p,f)+1<=len f -Index(p,f);
    then i1<=len f -Index(p,f) by A10,XREAL_0:def 2;
    then
A38: i1+1<=len L_Cut(f,p) by A32,XREAL_1:6;
    1<=Index(q,f) by A2,Th8;
    then
A39: 1<=Index(q,f)+1 by NAT_D:48;
A40: q in LSeg(f,Index(q,f)) by A2,Th9;
A41: Index(q,f)+1<=len f by A34,NAT_1:13;
A42: Index(q,f)-Index(p,f)+1+1<=len f -Index(p,f)+1 by A36,XREAL_1:6;
    then
A43: Index(q,f)-'Index(p,f)+1+1<=len <*p*>+len mid(f,Index(p,f)+1,len f)
    by A3,A18,XREAL_1:233;
    len L_Cut(f,p)>=Index(q,f)-'Index(p,f)+1+1 by A3,A32,A42,XREAL_1:233;
    then
    (L_Cut(f,p))/.(Index(q,f)-'Index(p,f)+1+1) = L_Cut(f,p).(Index(q,f)-'
    Index(p,f)+1+1) by FINSEQ_4:15,NAT_1:11
      .= (<*p*>^mid(f,Index(p,f)+1,len f)).(Index(q,f)-'Index(p,f)+1+1) by A31
,Def3
      .= mid(f,Index(p,f)+1,len f).(Index(q,f)-'Index(p,f)+1+1-len <*p*>) by
A15,A43,FINSEQ_6:108
      .= mid(f,Index(p,f)+1,len f).(Index(q,f)-'Index(p,f)+1+1-1) by
FINSEQ_1:40
      .= f.(Index(p,f)+1+(Index(q,f)-'Index(p,f)+1)-1) by A12,A8,A16,A37,
FINSEQ_6:122
      .= f.(Index(p,f)+1+(Index(q,f)-Index(p,f)+1)-1) by A3,XREAL_1:233
      .= f/.(Index(q,f)+1) by A39,A41,FINSEQ_4:15;
    then
    q in LSeg((L_Cut(f,p))/.((Index(q,f)-'Index(p,f)+1)), (L_Cut(f,p))/.(
    (Index(q,f)-'Index(p,f)+1+1))) by A17,A33,A35,A40,TOPREAL1:def 3;
    hence thesis by A38,NAT_1:11,SPPOL_2:15;
  end;
end;
