Article JORDAN3, MML version 4.99.1005

:: JORDAN3:th 15
theorem
for b1 being natural set
for b2, b3 being Relation-like Function-like FinSequence-like set
      st len b2 < b1 &
         (b1 <= (len b2) + len b3 or b1 <= len (b2 ^ b3))
   holds (b2 ^ b3) . b1 = b3 . (b1 - len b2);

:: JORDAN3:th 17
theorem
for b1 being non empty set
for b2 being set
for b3 being FinSequence of b1
      st 1 <= len b3
   holds (b3 ^ <*b2*>) . 1 = b3 . 1 &
    (b3 ^ <*b2*>) . 1 = b3 /. 1 &
    (<*b2*> ^ b3) . ((len b3) + 1) = b3 . len b3 &
    (<*b2*> ^ b3) . ((len b3) + 1) = b3 /. len b3;

:: JORDAN3:th 18
theorem
for b1 being Relation-like Function-like FinSequence-like set
      st len b1 = 1
   holds Rev b1 = b1;

:: JORDAN3:th 19
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being natural set holds
   len (b2 /^ b3) = (len b2) -' b3;

:: JORDAN3:th 21
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being natural set holds
(b2 /^ b3) | (b4 -' b3) = (b2 | b4) /^ b3;

:: JORDAN3:funcnot 1 => JORDAN3:func 1
definition
  let a1 be non empty set;
  let a2 be FinSequence of a1;
  let a3, a4 be natural set;
  func mid(A2,A3,A4) -> FinSequence of a1 equals
    (a2 /^ (a3 -' 1)) | ((a4 -' a3) + 1)
    if a3 <= a4
    otherwise Rev ((a2 /^ (a4 -' 1)) | ((a3 -' a4) + 1));
end;

:: JORDAN3:def 1
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being natural set holds
(b3 <= b4 implies mid(b2,b3,b4) = (b2 /^ (b3 -' 1)) | ((b4 -' b3) + 1)) &
 (b3 <= b4 or mid(b2,b3,b4) = Rev ((b2 /^ (b4 -' 1)) | ((b3 -' b4) + 1)));

:: JORDAN3:th 22
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT
      st 1 <= b3 & b3 <= len b2 & 1 <= b4 & b4 <= len b2
   holds Rev mid(b2,b3,b4) = mid(Rev b2,((len b2) -' b4) + 1,((len b2) -' b3) + 1);

:: JORDAN3:th 23
theorem
for b1 being non empty set
for b2, b3 being Element of NAT
for b4 being FinSequence of b1
      st 1 <= b3 & b3 + b2 <= len b4
   holds (b4 /^ b2) . b3 = b4 . (b3 + b2);

:: JORDAN3:th 24
theorem
for b1 being non empty set
for b2 being Element of NAT
for b3 being FinSequence of b1
      st 1 <= b2 & b2 <= len b3
   holds (Rev b3) . b2 = b3 . (((len b3) - b2) + 1);

:: JORDAN3:th 25
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being Element of NAT
      st 1 <= b3
   holds mid(b2,1,b3) = b2 | b3;

:: JORDAN3:th 26
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being Element of NAT
      st b3 <= len b2
   holds mid(b2,b3,len b2) = b2 /^ (b3 -' 1);

:: JORDAN3:th 27
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT
      st 1 <= b3 & b3 <= len b2 & 1 <= b4 & b4 <= len b2
   holds (mid(b2,b3,b4)) . 1 = b2 . b3 &
    (b3 <= b4 implies len mid(b2,b3,b4) = (b4 -' b3) + 1 &
     (for b5 being Element of NAT
           st 1 <= b5 & b5 <= len mid(b2,b3,b4)
        holds (mid(b2,b3,b4)) . b5 = b2 . ((b5 + b3) -' 1))) &
    (b3 <= b4 or len mid(b2,b3,b4) = (b3 -' b4) + 1 &
     (for b5 being Element of NAT
           st 1 <= b5 & b5 <= len mid(b2,b3,b4)
        holds (mid(b2,b3,b4)) . b5 = b2 . ((b3 -' b5) + 1)));

:: JORDAN3:th 28
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT holds
proj2 mid(b2,b3,b4) c= proj2 b2;

:: JORDAN3:th 29
theorem
for b1 being non empty set
for b2 being FinSequence of b1
      st 1 <= len b2
   holds mid(b2,1,len b2) = b2;

:: JORDAN3:th 30
theorem
for b1 being non empty set
for b2 being FinSequence of b1
      st 1 <= len b2
   holds mid(b2,len b2,1) = Rev b2;

:: JORDAN3:th 31
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4, b5 being Element of NAT
      st 1 <= b3 &
         b3 <= b4 &
         b4 <= len b2 &
         1 <= b5 &
         ((b4 -' b3) + 1 < b5 & (b4 - b3) + 1 < b5 implies b5 <= (b4 + 1) - b3)
   holds (mid(b2,b3,b4)) . b5 = b2 . ((b5 + b3) -' 1) &
    (mid(b2,b3,b4)) . b5 = b2 . ((b5 -' 1) + b3) &
    (mid(b2,b3,b4)) . b5 = b2 . ((b5 + b3) - 1) &
    (mid(b2,b3,b4)) . b5 = b2 . ((b5 - 1) + b3);

:: JORDAN3:th 32
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT
      st 1 <= b4 & b4 <= b3 & b3 <= len b2
   holds (mid(b2,1,b3)) . b4 = b2 . b4;

:: JORDAN3:th 33
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT
      st 1 <= b3 & b3 <= b4 & b4 <= len b2
   holds len mid(b2,b3,b4) <= len b2;

:: JORDAN3:th 34
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL b1
      st 2 <= len b2
   holds b2 . 1 in L~ b2 & b2 /. 1 in L~ b2 & b2 . len b2 in L~ b2 & b2 /. len b2 in L~ b2;

:: JORDAN3:th 35
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st (b1 `1 = b2 `1 or b1 `2 = b2 `2) &
         b3 in LSeg(b1,b2) &
         b4 in LSeg(b1,b2) &
         b3 `1 <> b4 `1
   holds b3 `2 = b4 `2;

:: JORDAN3:th 36
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st (b1 `1 = b2 `1 or b1 `2 = b2 `2) &
         LSeg(b3,b4) c= LSeg(b1,b2) &
         b3 `1 <> b4 `1
   holds b3 `2 = b4 `2;

:: JORDAN3:th 37
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 2 <= b2 & b1 is being_S-Seq
   holds b1 | b2 is being_S-Seq;

:: JORDAN3:th 38
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st b2 <= len b1 & 2 <= (len b1) -' b2 & b1 is being_S-Seq
   holds b1 /^ b2 is being_S-Seq;

:: JORDAN3:th 39
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st b1 is being_S-Seq & 1 <= b2 & b2 <= len b1 & 1 <= b3 & b3 <= len b1 & b2 <> b3
   holds mid(b1,b2,b3) is being_S-Seq;

:: JORDAN3:funcnot 2 => JORDAN3:func 2
definition
  let a1 be FinSequence of the carrier of TOP-REAL 2;
  let a2 be Element of the carrier of TOP-REAL 2;
  assume a2 in L~ a1;
  func Index(A2,A1) -> Element of NAT means
    ex b1 being non empty Element of bool NAT st
       it = min b1 &
        b1 = {b2 where b2 is Element of NAT: a2 in LSeg(a1,b2)};
end;

:: JORDAN3:def 2
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
   st b2 in L~ b1
for b3 being Element of NAT holds
      b3 = Index(b2,b1)
   iff
      ex b4 being non empty Element of bool NAT st
         b3 = min b4 &
          b4 = {b5 where b5 is Element of NAT: b2 in LSeg(b1,b5)};

:: JORDAN3:th 40
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of NAT
      st b2 in LSeg(b1,b3)
   holds Index(b2,b1) <= b3;

:: JORDAN3:th 41
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1
   holds 1 <= Index(b2,b1) & Index(b2,b1) < len b1;

:: JORDAN3:th 42
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1
   holds b2 in LSeg(b1,Index(b2,b1));

:: JORDAN3:th 43
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in LSeg(b1,1)
   holds Index(b2,b1) = 1;

:: JORDAN3:th 44
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st 2 <= len b1
   holds Index(b1 /. 1,b1) = 1;

:: JORDAN3:th 45
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
for b3 being natural set
      st b1 is being_S-Seq & 1 < b3 & b3 <= len b1 & b2 = b1 . b3
   holds (Index(b2,b1)) + 1 = b3;

:: JORDAN3:th 46
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of NAT
      st b1 is s.n.c. & b2 in LSeg(b1,b3) & b3 <> Index(b2,b1)
   holds b3 = (Index(b2,b1)) + 1;

:: JORDAN3:th 47
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of NAT
      st b1 is unfolded & b1 is s.n.c. & b3 + 1 <= len b1 & b2 in LSeg(b1,b3) & b2 <> b1 . b3
   holds b3 = Index(b2,b1);

:: JORDAN3:prednot 1 => JORDAN3:pred 1
definition
  let a1 be FinSequence of the carrier of TOP-REAL 2;
  let a2, a3 be Element of the carrier of TOP-REAL 2;
  pred A1 is_S-Seq_joining A2,A3 means
    a1 is being_S-Seq & a1 . 1 = a2 & a1 . len a1 = a3;
end;

:: JORDAN3:dfs 3
definiens
  let a1 be FinSequence of the carrier of TOP-REAL 2;
  let a2, a3 be Element of the carrier of TOP-REAL 2;
To prove
     a1 is_S-Seq_joining a2,a3
it is sufficient to prove
  thus a1 is being_S-Seq & a1 . 1 = a2 & a1 . len a1 = a3;

:: JORDAN3:def 3
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2 holds
   b1 is_S-Seq_joining b2,b3
iff
   b1 is being_S-Seq & b1 . 1 = b2 & b1 . len b1 = b3;

:: JORDAN3:th 48
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is_S-Seq_joining b2,b3
   holds Rev b1 is_S-Seq_joining b3,b2;

:: JORDAN3:th 49
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
for b4 being natural set
      st b3 in L~ b1 &
         b2 = <*b3*> ^ mid(b1,(Index(b3,b1)) + 1,len b1) &
         1 <= b4 &
         b4 + 1 <= len b2
   holds LSeg(b2,b4) c= LSeg(b1,((Index(b3,b1)) + b4) -' 1);

:: JORDAN3:th 50
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
      st b1 is being_S-Seq &
         b3 in L~ b1 &
         b3 <> b1 . ((Index(b3,b1)) + 1) &
         b2 = <*b3*> ^ mid(b1,(Index(b3,b1)) + 1,len b1)
   holds b2 is_S-Seq_joining b3,b1 /. len b1;

:: JORDAN3:th 51
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
for b4 being natural set
      st b3 in L~ b1 &
         1 <= b4 &
         b4 + 1 <= len b2 &
         b2 = (mid(b1,1,Index(b3,b1))) ^ <*b3*>
   holds LSeg(b2,b4) c= LSeg(b1,b4);

:: JORDAN3:th 52
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
      st b1 is being_S-Seq &
         b3 in L~ b1 &
         b3 <> b1 . 1 &
         b2 = (mid(b1,1,Index(b3,b1))) ^ <*b3*>
   holds b2 is_S-Seq_joining b1 /. 1,b3;

:: JORDAN3:funcnot 3 => JORDAN3:func 3
definition
  let a1 be FinSequence of the carrier of TOP-REAL 2;
  let a2 be Element of the carrier of TOP-REAL 2;
  func L_Cut(A1,A2) -> FinSequence of the carrier of TOP-REAL 2 equals
    <*a2*> ^ mid(a1,(Index(a2,a1)) + 1,len a1)
    if a2 <> a1 . ((Index(a2,a1)) + 1)
    otherwise mid(a1,(Index(a2,a1)) + 1,len a1);
end;

:: JORDAN3:def 4
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2 holds
   (b2 = b1 . ((Index(b2,b1)) + 1) or L_Cut(b1,b2) = <*b2*> ^ mid(b1,(Index(b2,b1)) + 1,len b1)) &
    (b2 = b1 . ((Index(b2,b1)) + 1) implies L_Cut(b1,b2) = mid(b1,(Index(b2,b1)) + 1,len b1));

:: JORDAN3:funcnot 4 => JORDAN3:func 4
definition
  let a1 be FinSequence of the carrier of TOP-REAL 2;
  let a2 be Element of the carrier of TOP-REAL 2;
  func R_Cut(A1,A2) -> FinSequence of the carrier of TOP-REAL 2 equals
    (mid(a1,1,Index(a2,a1))) ^ <*a2*>
    if a2 <> a1 . 1
    otherwise <*a2*>;
end;

:: JORDAN3:def 5
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2 holds
   (b2 = b1 . 1 or R_Cut(b1,b2) = (mid(b1,1,Index(b2,b1))) ^ <*b2*>) &
    (b2 = b1 . 1 implies R_Cut(b1,b2) = <*b2*>);

:: JORDAN3:th 53
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b1 is being_S-Seq & b2 in L~ b1 & b2 = b1 . ((Index(b2,b1)) + 1) & b2 <> b1 . len b1
   holds ((Index(b2,Rev b1)) + Index(b2,b1)) + 1 = len b1;

:: JORDAN3:th 54
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b1 is unfolded & b1 is s.n.c. & b2 in L~ b1 & b2 <> b1 . ((Index(b2,b1)) + 1)
   holds (Index(b2,Rev b1)) + Index(b2,b1) = len b1;

:: JORDAN3:th 55
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being Element of NAT
for b4 being Element of b1 holds
   (<*b4*> ^ b2) | (b3 + 1) = <*b4*> ^ (b2 | b3);

:: JORDAN3:th 56
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT
      st b3 < b4 & b3 in dom b2
   holds mid(b2,b3,b4) = <*b2 . b3*> ^ mid(b2,b3 + 1,b4);

:: JORDAN3:th 57
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b1 is being_S-Seq & b2 in L~ b1
   holds L_Cut(Rev b1,b2) = Rev R_Cut(b1,b2);

:: JORDAN3:th 58
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1
   holds (L_Cut(b1,b2)) . 1 = b2 &
    (for b3 being Element of NAT
          st 1 < b3 & b3 <= len L_Cut(b1,b2)
       holds (b2 = b1 . ((Index(b2,b1)) + 1) implies (L_Cut(b1,b2)) . b3 = b1 . ((Index(b2,b1)) + b3)) &
        (b2 = b1 . ((Index(b2,b1)) + 1) or (L_Cut(b1,b2)) . b3 = b1 . (((Index(b2,b1)) + b3) - 1)));

:: JORDAN3:th 59
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1
   holds (R_Cut(b1,b2)) . len R_Cut(b1,b2) = b2 &
    (for b3 being Element of NAT
          st 1 <= b3 & b3 <= Index(b2,b1)
       holds (R_Cut(b1,b2)) . b3 = b1 . b3);

:: JORDAN3:th 60
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1
   holds (b2 = b1 . 1 or len R_Cut(b1,b2) = (Index(b2,b1)) + 1) &
    (b2 = b1 . 1 implies len R_Cut(b1,b2) = Index(b2,b1));

:: JORDAN3:th 61
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1
   holds (b2 = b1 . ((Index(b2,b1)) + 1) implies len L_Cut(b1,b2) = (len b1) - Index(b2,b1)) &
    (b2 = b1 . ((Index(b2,b1)) + 1) or len L_Cut(b1,b2) = ((len b1) - Index(b2,b1)) + 1);

:: JORDAN3:prednot 2 => JORDAN3:pred 2
definition
  let a1, a2, a3, a4 be Element of the carrier of TOP-REAL 2;
  pred LE A3,A4,A1,A2 means
    a3 in LSeg(a1,a2) &
     a4 in LSeg(a1,a2) &
     (for b1, b2 being Element of REAL
           st 0 <= b1 &
              b1 <= 1 &
              a3 = ((1 - b1) * a1) + (b1 * a2) &
              0 <= b2 &
              b2 <= 1 &
              a4 = ((1 - b2) * a1) + (b2 * a2)
        holds b1 <= b2);
end;

:: JORDAN3:dfs 6
definiens
  let a1, a2, a3, a4 be Element of the carrier of TOP-REAL 2;
To prove
     LE a3,a4,a1,a2
it is sufficient to prove
  thus a3 in LSeg(a1,a2) &
     a4 in LSeg(a1,a2) &
     (for b1, b2 being Element of REAL
           st 0 <= b1 &
              b1 <= 1 &
              a3 = ((1 - b1) * a1) + (b1 * a2) &
              0 <= b2 &
              b2 <= 1 &
              a4 = ((1 - b2) * a1) + (b2 * a2)
        holds b1 <= b2);

:: JORDAN3:def 6
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2 holds
   LE b3,b4,b1,b2
iff
   b3 in LSeg(b1,b2) &
    b4 in LSeg(b1,b2) &
    (for b5, b6 being Element of REAL
          st 0 <= b5 &
             b5 <= 1 &
             b3 = ((1 - b5) * b1) + (b5 * b2) &
             0 <= b6 &
             b6 <= 1 &
             b4 = ((1 - b6) * b1) + (b6 * b2)
       holds b5 <= b6);

:: JORDAN3:prednot 3 => JORDAN3:pred 3
definition
  let a1, a2, a3, a4 be Element of the carrier of TOP-REAL 2;
  pred LT A3,A4,A1,A2 means
    LE a3,a4,a1,a2 & a3 <> a4;
end;

:: JORDAN3:dfs 7
definiens
  let a1, a2, a3, a4 be Element of the carrier of TOP-REAL 2;
To prove
     LT a3,a4,a1,a2
it is sufficient to prove
  thus LE a3,a4,a1,a2 & a3 <> a4;

:: JORDAN3:def 7
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2 holds
   LT b3,b4,b1,b2
iff
   LE b3,b4,b1,b2 & b3 <> b4;

:: JORDAN3:th 62
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st LE b3,b4,b1,b2 & LE b4,b3,b1,b2
   holds b3 = b4;

:: JORDAN3:th 63
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b3 in LSeg(b1,b2) & b4 in LSeg(b1,b2) & b1 <> b2
   holds (LE b3,b4,b1,b2 or LT b4,b3,b1,b2) & (LE b3,b4,b1,b2 implies not LT b4,b3,b1,b2);

:: JORDAN3:th 64
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1 & b3 in L~ b1 & Index(b2,b1) < Index(b3,b1)
   holds b3 in L~ L_Cut(b1,b2);

:: JORDAN3:th 65
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st LE b1,b2,b3,b4
   holds b2 in LSeg(b1,b4) & b1 in LSeg(b3,b2);

:: JORDAN3:th 66
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1 &
         b3 in L~ b1 &
         b2 <> b3 &
         Index(b2,b1) = Index(b3,b1) &
         LE b2,b3,b1 /. Index(b2,b1),b1 /. ((Index(b2,b1)) + 1)
   holds b3 in L~ L_Cut(b1,b2);

:: JORDAN3:funcnot 5 => JORDAN3:func 5
definition
  let a1 be FinSequence of the carrier of TOP-REAL 2;
  let a2, a3 be Element of the carrier of TOP-REAL 2;
  func B_Cut(A1,A2,A3) -> FinSequence of the carrier of TOP-REAL 2 equals
    R_Cut(L_Cut(a1,a2),a3)
    if (a2 in L~ a1 & a3 in L~ a1 & Index(a2,a1) < Index(a3,a1) or Index(a2,a1) = Index(a3,a1) &
     LE a2,a3,a1 /. Index(a2,a1),a1 /. ((Index(a2,a1)) + 1))
    otherwise Rev R_Cut(L_Cut(a1,a3),a2);
end;

:: JORDAN3:def 8
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2 holds
((b2 in L~ b1 & b3 in L~ b1 implies Index(b3,b1) <= Index(b2,b1)) &
  (Index(b2,b1) = Index(b3,b1) implies not LE b2,b3,b1 /. Index(b2,b1),b1 /. ((Index(b2,b1)) + 1)) or B_Cut(b1,b2,b3) = R_Cut(L_Cut(b1,b2),b3)) &
 ((b2 in L~ b1 & b3 in L~ b1 implies Index(b3,b1) <= Index(b2,b1)) &
  (Index(b2,b1) = Index(b3,b1) implies not LE b2,b3,b1 /. Index(b2,b1),b1 /. ((Index(b2,b1)) + 1)) implies B_Cut(b1,b2,b3) = Rev R_Cut(L_Cut(b1,b3),b2));

:: JORDAN3:th 67
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b1 is being_S-Seq & b2 in L~ b1 & b2 <> b1 . 1
   holds R_Cut(b1,b2) is_S-Seq_joining b1 /. 1,b2;

:: JORDAN3:th 68
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b1 is being_S-Seq & b2 in L~ b1 & b2 <> b1 . len b1
   holds L_Cut(b1,b2) is_S-Seq_joining b2,b1 /. len b1;

:: JORDAN3:th 69
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b1 is being_S-Seq & b2 in L~ b1 & b2 <> b1 . len b1
   holds L_Cut(b1,b2) is being_S-Seq;

:: JORDAN3:th 70
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b1 is being_S-Seq & b2 in L~ b1 & b2 <> b1 . 1
   holds R_Cut(b1,b2) is being_S-Seq;

:: JORDAN3:th 71
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is being_S-Seq & b2 in L~ b1 & b3 in L~ b1 & b2 <> b3
   holds B_Cut(b1,b2,b3) is_S-Seq_joining b2,b3;

:: JORDAN3:th 72
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is being_S-Seq & b2 in L~ b1 & b3 in L~ b1 & b2 <> b3
   holds B_Cut(b1,b2,b3) is being_S-Seq;

:: JORDAN3:th 73
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
      st b1 . len b1 = b2 . 1 &
         b1 is being_S-Seq &
         b2 is being_S-Seq &
         (L~ b1) /\ L~ b2 = {b2 . 1}
   holds b1 ^ mid(b2,2,len b2) is being_S-Seq;

:: JORDAN3:th 74
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
      st b1 . len b1 = b2 . 1 &
         b1 is being_S-Seq &
         b2 is being_S-Seq &
         (L~ b1) /\ L~ b2 = {b2 . 1}
   holds b1 ^ mid(b2,2,len b2) is_S-Seq_joining b1 /. 1,b2 /. len b2;

:: JORDAN3:th 75
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   L~ (b1 /^ b2) c= L~ b1;

:: JORDAN3:th 76
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1
   holds L~ R_Cut(b1,b2) c= L~ b1;

:: JORDAN3:th 77
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1
   holds L~ L_Cut(b1,b2) c= L~ b1;

:: JORDAN3:th 78
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
      st b1 . len b1 = b2 . 1 &
         b3 in L~ b1 &
         b1 is being_S-Seq &
         b2 is being_S-Seq &
         (L~ b1) /\ L~ b2 = {b2 . 1} &
         b3 <> b1 . len b1
   holds (L_Cut(b1,b3)) ^ mid(b2,2,len b2) is_S-Seq_joining b3,b2 /. len b2;

:: JORDAN3:th 79
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
      st b1 . len b1 = b2 . 1 &
         b3 in L~ b1 &
         b1 is being_S-Seq &
         b2 is being_S-Seq &
         (L~ b1) /\ L~ b2 = {b2 . 1} &
         b3 <> b1 . len b1
   holds (L_Cut(b1,b3)) ^ mid(b2,2,len b2) is being_S-Seq;

:: JORDAN3:th 80
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
      st b1 . len b1 = b2 . 1 &
         b1 is being_S-Seq &
         b2 is being_S-Seq &
         (L~ b1) /\ L~ b2 = {b2 . 1}
   holds (mid(b1,1,(len b1) -' 1)) ^ b2 is being_S-Seq;

:: JORDAN3:th 81
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
      st b1 . len b1 = b2 . 1 &
         b1 is being_S-Seq &
         b2 is being_S-Seq &
         (L~ b1) /\ L~ b2 = {b2 . 1}
   holds (mid(b1,1,(len b1) -' 1)) ^ b2 is_S-Seq_joining b1 /. 1,b2 /. len b2;

:: JORDAN3:th 82
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
      st b1 . len b1 = b2 . 1 &
         b3 in L~ b2 &
         b1 is being_S-Seq &
         b2 is being_S-Seq &
         (L~ b1) /\ L~ b2 = {b2 . 1} &
         b3 <> b2 . 1
   holds (mid(b1,1,(len b1) -' 1)) ^ R_Cut(b2,b3) is_S-Seq_joining b1 /. 1,b3;

:: JORDAN3:th 83
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
      st b1 . len b1 = b2 . 1 &
         b3 in L~ b2 &
         b1 is being_S-Seq &
         b2 is being_S-Seq &
         (L~ b1) /\ L~ b2 = {b2 . 1} &
         b3 <> b2 . 1
   holds (mid(b1,1,(len b1) -' 1)) ^ R_Cut(b2,b3) is being_S-Seq;