Article JORDAN4, MML version 4.99.1005

:: JORDAN4:th 6
theorem
for b1, b2 being natural set
      st 0 < b2 & b2 < b1 & b1 < b2 + b2
   holds b1 mod b2 <> 0;

:: JORDAN4:th 7
theorem
for b1, b2 being natural set
      st 0 < b2 & b2 <= b1 & b1 < b2 + b2
   holds b1 mod b2 = b1 - b2 & b1 mod b2 = b1 -' b2;

:: JORDAN4:th 8
theorem
for b1, b2 being natural set holds
(b2 + b2) mod b2 = 0;

:: JORDAN4:th 9
theorem
for b1, b2 being natural set
      st 0 < b2 & b2 <= b1 & b2 mod b1 = 0
   holds b2 = b1;

:: JORDAN4:th 14
theorem
for b1 being non empty set
for b2 being FinSequence of b1
      st b2 is circular(b1) & 1 <= len b2
   holds b2 . 1 = b2 . len b2;

:: JORDAN4:th 18
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being Element of NAT
      st b3 < len b2
   holds (b2 /^ b3) . len (b2 /^ b3) = b2 . len b2 &
    (b2 /^ b3) /. len (b2 /^ b3) = b2 /. len b2;

:: JORDAN4:th 19
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st b1 is being_S-Seq & b2 + 1 < len b1
   holds b1 /^ b2 is being_S-Seq;

:: JORDAN4:th 20
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 len mid(b2,b4,b3) = (b3 -' b4) + 1;

:: JORDAN4:th 21
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 len mid(b2,b3,b4) = (b3 -' b4) + 1;

:: JORDAN4:th 22
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
   holds (mid(b2,b3,b4)) . len mid(b2,b3,b4) = b2 . b4;

:: JORDAN4:th 23
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 <= len b2 & 1 <= b4 & b4 <= len b2
   holds (mid(b2,b3,b4)) . len mid(b2,b3,b4) = b2 . b4;

:: JORDAN4:th 24
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4, b5 being Element of NAT
      st 1 <= b4 & b4 <= b3 & b3 <= len b2 & 1 <= b5 & b5 <= (b3 -' b4) + 1
   holds (mid(b2,b3,b4)) . b5 = b2 . ((b3 -' b5) + 1);

:: JORDAN4:th 25
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being FinSequence of b2
for b4, b5 being Element of NAT
      st 1 <= b5 & b5 <= b4 & b4 <= len b3 & 1 <= b1 & b1 <= (b4 -' b5) + 1
   holds (mid(b3,b4,b5)) . b1 = (mid(b3,b5,b4)) . ((((b4 - b5) + 1) - b1) + 1) &
    (((b4 - b5) + 1) - b1) + 1 = (((b4 -' b5) + 1) -' b1) + 1;

:: JORDAN4:th 26
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being FinSequence of b2
for b4, b5 being Element of NAT
      st 1 <= b4 & b4 <= b5 & b5 <= len b3 & 1 <= b1 & b1 <= (b5 -' b4) + 1
   holds (mid(b3,b4,b5)) . b1 = (mid(b3,b5,b4)) . ((((b5 - b4) + 1) - b1) + 1) &
    (((b5 - b4) + 1) - b1) + 1 = (((b5 -' b4) + 1) -' b1) + 1;

:: JORDAN4:th 27
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being Element of NAT
      st 1 <= b3 & b3 <= len b2
   holds mid(b2,b3,b3) = <*b2 /. b3*> &
    len mid(b2,b3,b3) = 1;

:: JORDAN4:th 28
theorem
for b1 being non empty set
for b2 being FinSequence of b1 holds
   mid(b2,0,0) = b2 | 1;

:: JORDAN4:th 29
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being Element of NAT
      st len b2 < b3
   holds mid(b2,b3,b3) = <*> b1;

:: JORDAN4:th 30
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT holds
mid(b2,b3,b4) = Rev mid(b2,b4,b3);

:: JORDAN4:th 31
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3, b4 being Element of NAT
      st 1 <= b2 & b2 < b3 & b3 <= len b1 & 1 <= b4 & b4 < (b3 -' b2) + 1
   holds LSeg(mid(b1,b2,b3),b4) = LSeg(b1,(b4 + b2) -' 1);

:: JORDAN4:th 32
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3, b4 being Element of NAT
      st 1 <= b2 & b2 < b3 & b3 <= len b1 & 1 <= b4 & b4 < (b3 -' b2) + 1
   holds LSeg(mid(b1,b3,b2),b4) = LSeg(b1,b3 -' b4);

:: JORDAN4:funcnot 1 => JORDAN4:func 1
definition
  let a1 be Element of NAT;
  let a2 be Relation-like Function-like FinSequence-like set;
  func S_Drop(A1,A2) -> Element of NAT equals
    a1 mod ((len a2) -' 1)
    if a1 mod ((len a2) -' 1) <> 0
    otherwise (len a2) -' 1;
end;

:: JORDAN4:def 1
theorem
for b1 being Element of NAT
for b2 being Relation-like Function-like FinSequence-like set holds
   (b1 mod ((len b2) -' 1) = 0 or S_Drop(b1,b2) = b1 mod ((len b2) -' 1)) &
    (b1 mod ((len b2) -' 1) = 0 implies S_Drop(b1,b2) = (len b2) -' 1);

:: JORDAN4:th 33
theorem
for b1 being Relation-like Function-like FinSequence-like set holds
   S_Drop((len b1) -' 1,b1) = (len b1) -' 1;

:: JORDAN4:th 34
theorem
for b1 being Element of NAT
for b2 being Relation-like Function-like FinSequence-like set
      st 1 <= b1 & b1 <= (len b2) -' 1
   holds S_Drop(b1,b2) = b1;

:: JORDAN4:th 35
theorem
for b1 being Element of NAT
for b2 being Relation-like Function-like FinSequence-like set holds
   S_Drop(b1,b2) = S_Drop(b1 + ((len b2) -' 1),b2) &
    S_Drop(b1,b2) = S_Drop(((len b2) -' 1) + b1,b2);

:: JORDAN4:prednot 1 => JORDAN4:pred 1
definition
  let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
  let a2 be FinSequence of the carrier of TOP-REAL 2;
  let a3, a4 be Element of NAT;
  pred A2 is_a_part>_of A1,A3,A4 means
    1 <= a3 &
     a3 + 1 <= len a1 &
     1 <= a4 &
     a4 + 1 <= len a1 &
     a2 . len a2 = a1 . a4 &
     1 <= len a2 &
     len a2 < len a1 &
     (for b1 being natural set
           st 1 <= b1 & b1 <= len a2
        holds a2 . b1 = a1 . S_Drop((a3 + b1) -' 1,a1));
end;

:: JORDAN4:dfs 2
definiens
  let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
  let a2 be FinSequence of the carrier of TOP-REAL 2;
  let a3, a4 be Element of NAT;
To prove
     a2 is_a_part>_of a1,a3,a4
it is sufficient to prove
  thus 1 <= a3 &
     a3 + 1 <= len a1 &
     1 <= a4 &
     a4 + 1 <= len a1 &
     a2 . len a2 = a1 . a4 &
     1 <= len a2 &
     len a2 < len a1 &
     (for b1 being natural set
           st 1 <= b1 & b1 <= len a2
        holds a2 . b1 = a1 . S_Drop((a3 + b1) -' 1,a1));

:: JORDAN4:def 2
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT holds
   b2 is_a_part>_of b1,b3,b4
iff
   1 <= b3 &
    b3 + 1 <= len b1 &
    1 <= b4 &
    b4 + 1 <= len b1 &
    b2 . len b2 = b1 . b4 &
    1 <= len b2 &
    len b2 < len b1 &
    (for b5 being natural set
          st 1 <= b5 & b5 <= len b2
       holds b2 . b5 = b1 . S_Drop((b3 + b5) -' 1,b1));

:: JORDAN4:prednot 2 => JORDAN4:pred 2
definition
  let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
  let a2 be FinSequence of the carrier of TOP-REAL 2;
  let a3, a4 be Element of NAT;
  pred A2 is_a_part<_of A1,A3,A4 means
    1 <= a3 &
     a3 + 1 <= len a1 &
     1 <= a4 &
     a4 + 1 <= len a1 &
     a2 . len a2 = a1 . a4 &
     1 <= len a2 &
     len a2 < len a1 &
     (for b1 being natural set
           st 1 <= b1 & b1 <= len a2
        holds a2 . b1 = a1 . S_Drop(((len a1) + a3) -' b1,a1));
end;

:: JORDAN4:dfs 3
definiens
  let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
  let a2 be FinSequence of the carrier of TOP-REAL 2;
  let a3, a4 be Element of NAT;
To prove
     a2 is_a_part<_of a1,a3,a4
it is sufficient to prove
  thus 1 <= a3 &
     a3 + 1 <= len a1 &
     1 <= a4 &
     a4 + 1 <= len a1 &
     a2 . len a2 = a1 . a4 &
     1 <= len a2 &
     len a2 < len a1 &
     (for b1 being natural set
           st 1 <= b1 & b1 <= len a2
        holds a2 . b1 = a1 . S_Drop(((len a1) + a3) -' b1,a1));

:: JORDAN4:def 3
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT holds
   b2 is_a_part<_of b1,b3,b4
iff
   1 <= b3 &
    b3 + 1 <= len b1 &
    1 <= b4 &
    b4 + 1 <= len b1 &
    b2 . len b2 = b1 . b4 &
    1 <= len b2 &
    len b2 < len b1 &
    (for b5 being natural set
          st 1 <= b5 & b5 <= len b2
       holds b2 . b5 = b1 . S_Drop(((len b1) + b3) -' b5,b1));

:: JORDAN4:prednot 3 => JORDAN4:pred 3
definition
  let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
  let a2 be FinSequence of the carrier of TOP-REAL 2;
  let a3, a4 be Element of NAT;
  pred A2 is_a_part_of A1,A3,A4 means
    (not a2 is_a_part>_of a1,a3,a4) implies a2 is_a_part<_of a1,a3,a4;
end;

:: JORDAN4:dfs 4
definiens
  let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
  let a2 be FinSequence of the carrier of TOP-REAL 2;
  let a3, a4 be Element of NAT;
To prove
     a2 is_a_part_of a1,a3,a4
it is sufficient to prove
  thus (not a2 is_a_part>_of a1,a3,a4) implies a2 is_a_part<_of a1,a3,a4;

:: JORDAN4:def 4
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT holds
   b2 is_a_part_of b1,b3,b4
iff
   (b2 is_a_part>_of b1,b3,b4 or b2 is_a_part<_of b1,b3,b4);

:: JORDAN4:th 36
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
      st b2 is_a_part_of b1,b3,b4
   holds 1 <= b3 &
    b3 + 1 <= len b1 &
    1 <= b4 &
    b4 + 1 <= len b1 &
    b2 . len b2 = b1 . b4 &
    1 <= len b2 &
    len b2 < len b1 &
    (for b5 being Element of NAT
          st 1 <= b5 & b5 <= len b2
       holds b2 . b5 = b1 . S_Drop((b3 + b5) -' 1,b1) or for b5 being Element of NAT
          st 1 <= b5 & b5 <= len b2
       holds b2 . b5 = b1 . S_Drop(((len b1) + b3) -' b5,b1));

:: JORDAN4:th 37
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
      st b2 is_a_part>_of b1,b3,b4 & b3 <= b4
   holds len b2 = (b4 -' b3) + 1 & b2 = mid(b1,b3,b4);

:: JORDAN4:th 38
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
      st b2 is_a_part>_of b1,b3,b4 & b4 < b3
   holds len b2 = ((len b1) + b4) -' b3 &
    b2 = (mid(b1,b3,(len b1) -' 1)) ^ (b1 | b4) &
    b2 = (mid(b1,b3,(len b1) -' 1)) ^ mid(b1,1,b4);

:: JORDAN4:th 39
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
      st b2 is_a_part<_of b1,b3,b4 & b4 <= b3
   holds len b2 = (b3 -' b4) + 1 & b2 = mid(b1,b3,b4);

:: JORDAN4:th 40
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
      st b2 is_a_part<_of b1,b3,b4 & b3 < b4
   holds len b2 = ((len b1) + b3) -' b4 &
    b2 = (mid(b1,b3,1)) ^ mid(b1,(len b1) -' 1,b4);

:: JORDAN4:th 41
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
      st b2 is_a_part>_of b1,b3,b4
   holds Rev b2 is_a_part<_of b1,b4,b3;

:: JORDAN4:th 42
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
      st b2 is_a_part<_of b1,b3,b4
   holds Rev b2 is_a_part>_of b1,b4,b3;

:: JORDAN4:th 43
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st 1 <= b2 & b2 <= b3 & b3 < len b1
   holds mid(b1,b2,b3) is_a_part>_of b1,b2,b3;

:: JORDAN4:th 44
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st 1 <= b2 & b2 <= b3 & b3 < len b1
   holds mid(b1,b3,b2) is_a_part<_of b1,b3,b2;

:: JORDAN4:th 45
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st 1 <= b3 & b3 < b2 & b2 < len b1
   holds (mid(b1,b2,(len b1) -' 1)) ^ mid(b1,1,b3) is_a_part>_of b1,b2,b3;

:: JORDAN4:th 46
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st 1 <= b2 & b2 < b3 & b3 < len b1
   holds (mid(b1,b2,1)) ^ mid(b1,(len b1) -' 1,b3) is_a_part<_of b1,b2,b3;

:: JORDAN4:th 47
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st 1 <= b2 & b2 <= len b1 & 1 <= b3 & b3 <= len b1
   holds L~ mid(b1,b2,b3) c= L~ b1;

:: JORDAN4:th 48
theorem
for b1 being non empty set
for b2 being FinSequence of b1 holds
      b2 is one-to-one
   iff
      for b3, b4 being Element of NAT
            st 1 <= b3 &
               b3 <= len b2 &
               1 <= b4 &
               b4 <= len b2 &
               (b2 . b3 = b2 . b4 or b2 /. b3 = b2 /. b4)
         holds b3 = b4;

:: JORDAN4:th 49
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 1 < b2 & b2 + 1 <= len b1
   holds b1 | b2 is being_S-Seq;

:: JORDAN4:th 50
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 1 <= b2 & b2 + 1 < len b1
   holds b1 /^ b2 is being_S-Seq;

:: JORDAN4:th 51
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st 1 <= b2 & b2 < b3 & b3 + 1 <= len b1
   holds mid(b1,b2,b3) is being_S-Seq;

:: JORDAN4:th 52
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st 1 < b2 & b2 < b3 & b3 <= len b1
   holds mid(b1,b2,b3) is being_S-Seq;

:: JORDAN4:th 53
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 in LSeg(b2,b3) & b1 in LSeg(b2,b4) & b2 <> b1 & not b3 in LSeg(b2,b4)
   holds b4 in LSeg(b2,b3);

:: JORDAN4:th 54
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   (LSeg(b1,1)) /\ LSeg(b1,(len b1) -' 1) = {b1 . 1};

:: JORDAN4:th 55
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
for b4, b5 being FinSequence of the carrier of TOP-REAL 2
      st 1 <= b2 &
         b2 < b3 &
         b3 < len b1 &
         b4 = mid(b1,b2,b3) &
         b5 = (mid(b1,b2,1)) ^ mid(b1,(len b1) -' 1,b3)
   holds (L~ b4) /\ L~ b5 = {b1 . b2,b1 . b3} &
    (L~ b4) \/ L~ b5 = L~ b1;

:: JORDAN4:th 56
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
      st b2 is_a_part>_of b1,b3,b4 & b3 < b4
   holds L~ b2 is_S-P_arc_joining b1 /. b3,b1 /. b4;

:: JORDAN4:th 57
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
      st b2 is_a_part<_of b1,b3,b4 & b4 < b3
   holds L~ b2 is_S-P_arc_joining b1 /. b3,b1 /. b4;

:: JORDAN4:th 58
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
      st b2 is_a_part>_of b1,b3,b4 & b3 <> b4
   holds L~ b2 is_S-P_arc_joining b1 /. b3,b1 /. b4;

:: JORDAN4:th 59
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
      st b2 is_a_part<_of b1,b3,b4 & b3 <> b4
   holds L~ b2 is_S-P_arc_joining b1 /. b3,b1 /. b4;

:: JORDAN4:th 60
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
      st b2 is_a_part_of b1,b3,b4 & b3 <> b4
   holds L~ b2 is_S-P_arc_joining b1 /. b3,b1 /. b4;

:: JORDAN4:th 61
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
      st b2 is_a_part_of b1,b3,b4 & b2 . 1 <> b2 . len b2
   holds L~ b2 is_S-P_arc_joining b1 /. b3,b1 /. b4;

:: JORDAN4:th 62
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st 1 <= b2 & b2 + 1 <= len b1 & 1 <= b3 & b3 + 1 <= len b1 & b2 <> b3
   holds ex b4, b5 being FinSequence of the carrier of TOP-REAL 2 st
      b4 is_a_part_of b1,b2,b3 &
       b5 is_a_part_of b1,b2,b3 &
       (L~ b4) /\ L~ b5 = {b1 . b2,b1 . b3} &
       (L~ b4) \/ L~ b5 = L~ b1 &
       L~ b4 is_S-P_arc_joining b1 /. b2,b1 /. b3 &
       L~ b5 is_S-P_arc_joining b1 /. b2,b1 /. b3 &
       (for b6 being FinSequence of the carrier of TOP-REAL 2
             st b6 is_a_part_of b1,b2,b3 & b6 <> b4
          holds b6 = b5);

:: JORDAN4:th 63
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being non empty Element of bool the carrier of TOP-REAL 2
      st b2 = L~ b1
   holds b2 is being_simple_closed_curve;

:: JORDAN4:th 64
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b4, b5 being FinSequence of the carrier of TOP-REAL 2
      st b4 is_a_part>_of b3,b1,b2 & b5 is_a_part>_of b3,b1,b2
   holds b4 = b5;

:: JORDAN4:th 65
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b4, b5 being FinSequence of the carrier of TOP-REAL 2
      st b4 is_a_part<_of b3,b1,b2 & b5 is_a_part<_of b3,b1,b2
   holds b4 = b5;

:: JORDAN4:th 66
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b4, b5 being FinSequence of the carrier of TOP-REAL 2
      st b1 <> b2 & b4 is_a_part>_of b3,b1,b2 & b5 is_a_part<_of b3,b1,b2
   holds b4 . 2 <> b5 . 2;

:: JORDAN4:th 67
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b4, b5 being FinSequence of the carrier of TOP-REAL 2
      st b1 <> b2 & b4 is_a_part_of b3,b1,b2 & b5 is_a_part_of b3,b1,b2 & b4 . 2 = b5 . 2
   holds b4 = b5;

:: JORDAN4:funcnot 2 => JORDAN4:func 2
definition
  let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
  let a2, a3 be Element of NAT;
  assume 1 <= a2 & a2 + 1 <= len a1 & 1 <= a3 & a3 + 1 <= len a1 & a2 <> a3;
  func Lower(A1,A2,A3) -> FinSequence of the carrier of TOP-REAL 2 means
    it is_a_part_of a1,a2,a3 &
     ((a1 /. a2) `1 <= (a1 /. (a2 + 1)) `1 &
      (a1 /. a2) `2 <= (a1 /. (a2 + 1)) `2 or it . 2 = a1 . (a2 + 1)) &
     ((a1 /. a2) `1 <= (a1 /. (a2 + 1)) `1 &
      (a1 /. a2) `2 <= (a1 /. (a2 + 1)) `2 implies it . 2 = a1 . S_Drop(a2 -' 1,a1));
end;

:: JORDAN4:def 5
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
   st 1 <= b2 & b2 + 1 <= len b1 & 1 <= b3 & b3 + 1 <= len b1 & b2 <> b3
for b4 being FinSequence of the carrier of TOP-REAL 2 holds
      b4 = Lower(b1,b2,b3)
   iff
      b4 is_a_part_of b1,b2,b3 &
       ((b1 /. b2) `1 <= (b1 /. (b2 + 1)) `1 &
        (b1 /. b2) `2 <= (b1 /. (b2 + 1)) `2 or b4 . 2 = b1 . (b2 + 1)) &
       ((b1 /. b2) `1 <= (b1 /. (b2 + 1)) `1 &
        (b1 /. b2) `2 <= (b1 /. (b2 + 1)) `2 implies b4 . 2 = b1 . S_Drop(b2 -' 1,b1));

:: JORDAN4:funcnot 3 => JORDAN4:func 3
definition
  let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
  let a2, a3 be Element of NAT;
  assume 1 <= a2 & a2 + 1 <= len a1 & 1 <= a3 & a3 + 1 <= len a1 & a2 <> a3;
  func Upper(A1,A2,A3) -> FinSequence of the carrier of TOP-REAL 2 means
    it is_a_part_of a1,a2,a3 &
     ((a1 /. (a2 + 1)) `1 <= (a1 /. a2) `1 &
      (a1 /. (a2 + 1)) `2 <= (a1 /. a2) `2 or it . 2 = a1 . (a2 + 1)) &
     ((a1 /. (a2 + 1)) `1 <= (a1 /. a2) `1 &
      (a1 /. (a2 + 1)) `2 <= (a1 /. a2) `2 implies it . 2 = a1 . S_Drop(a2 -' 1,a1));
end;

:: JORDAN4:def 6
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
   st 1 <= b2 & b2 + 1 <= len b1 & 1 <= b3 & b3 + 1 <= len b1 & b2 <> b3
for b4 being FinSequence of the carrier of TOP-REAL 2 holds
      b4 = Upper(b1,b2,b3)
   iff
      b4 is_a_part_of b1,b2,b3 &
       ((b1 /. (b2 + 1)) `1 <= (b1 /. b2) `1 &
        (b1 /. (b2 + 1)) `2 <= (b1 /. b2) `2 or b4 . 2 = b1 . (b2 + 1)) &
       ((b1 /. (b2 + 1)) `1 <= (b1 /. b2) `1 &
        (b1 /. (b2 + 1)) `2 <= (b1 /. b2) `2 implies b4 . 2 = b1 . S_Drop(b2 -' 1,b1));