Article JORDAN5C, MML version 4.99.1005

:: JORDAN5C:th 1
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4, b5 being Element of the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b7 being Element of REAL
   st b5 in b1 &
      b5 in b2 &
      b6 . b7 = b5 &
      b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) &
      b6 . 0 = b3 &
      b6 . 1 = b4 &
      0 <= b7 &
      b7 <= 1 &
      (for b8 being Element of REAL
            st 0 <= b8 & b8 < b7
         holds not b6 . b8 in b2)
for b8 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b9 being Element of REAL
   st b8 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) & b8 . 0 = b3 & b8 . 1 = b4 & b8 . b9 = b5 & 0 <= b9 & b9 <= 1
for b10 being Element of REAL
      st 0 <= b10 & b10 < b9
   holds not b8 . b10 in b2;

:: JORDAN5C:funcnot 1 => JORDAN5C:func 1
definition
  let a1, a2 be Element of bool the carrier of TOP-REAL 2;
  let a3, a4 be Element of the carrier of TOP-REAL 2;
  assume a1 meets a2 & a1 /\ a2 is closed(TOP-REAL 2) & a1 is_an_arc_of a3,a4;
  func First_Point(A1,A3,A4,A2) -> Element of the carrier of TOP-REAL 2 means
    it in a1 /\ a2 &
     (for b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | a1
     for b2 being Element of REAL
        st b1 is being_homeomorphism(I[01], (TOP-REAL 2) | a1) & b1 . 0 = a3 & b1 . 1 = a4 & b1 . b2 = it & 0 <= b2 & b2 <= 1
     for b3 being Element of REAL
           st 0 <= b3 & b3 < b2
        holds not b1 . b3 in a2);
end;

:: JORDAN5C:def 1
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
   st b1 meets b2 & b1 /\ b2 is closed(TOP-REAL 2) & b1 is_an_arc_of b3,b4
for b5 being Element of the carrier of TOP-REAL 2 holds
      b5 = First_Point(b1,b3,b4,b2)
   iff
      b5 in b1 /\ b2 &
       (for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
       for b7 being Element of REAL
          st b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) & b6 . 0 = b3 & b6 . 1 = b4 & b6 . b7 = b5 & 0 <= b7 & b7 <= 1
       for b8 being Element of REAL
             st 0 <= b8 & b8 < b7
          holds not b6 . b8 in b2);

:: JORDAN5C:th 2
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b3 in b1 & b1 is_an_arc_of b4,b5 & b2 = {b3}
   holds First_Point(b1,b4,b5,b2) = b3;

:: JORDAN5C:th 3
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b3 in b2 & b1 /\ b2 is closed(TOP-REAL 2) & b1 is_an_arc_of b3,b4
   holds First_Point(b1,b3,b4,b2) = b3;

:: JORDAN5C:th 4
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4, b5 being Element of the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b7 being Element of REAL
   st b5 in b1 &
      b5 in b2 &
      b6 . b7 = b5 &
      b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) &
      b6 . 0 = b3 &
      b6 . 1 = b4 &
      0 <= b7 &
      b7 <= 1 &
      (for b8 being Element of REAL
            st b8 <= 1 & b7 < b8
         holds not b6 . b8 in b2)
for b8 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b9 being Element of REAL
   st b8 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) & b8 . 0 = b3 & b8 . 1 = b4 & b8 . b9 = b5 & 0 <= b9 & b9 <= 1
for b10 being Element of REAL
      st b10 <= 1 & b9 < b10
   holds not b8 . b10 in b2;

:: JORDAN5C:funcnot 2 => JORDAN5C:func 2
definition
  let a1, a2 be Element of bool the carrier of TOP-REAL 2;
  let a3, a4 be Element of the carrier of TOP-REAL 2;
  assume a1 meets a2 & a1 /\ a2 is closed(TOP-REAL 2) & a1 is_an_arc_of a3,a4;
  func Last_Point(A1,A3,A4,A2) -> Element of the carrier of TOP-REAL 2 means
    it in a1 /\ a2 &
     (for b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | a1
     for b2 being Element of REAL
        st b1 is being_homeomorphism(I[01], (TOP-REAL 2) | a1) & b1 . 0 = a3 & b1 . 1 = a4 & b1 . b2 = it & 0 <= b2 & b2 <= 1
     for b3 being Element of REAL
           st b3 <= 1 & b2 < b3
        holds not b1 . b3 in a2);
end;

:: JORDAN5C:def 2
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
   st b1 meets b2 & b1 /\ b2 is closed(TOP-REAL 2) & b1 is_an_arc_of b3,b4
for b5 being Element of the carrier of TOP-REAL 2 holds
      b5 = Last_Point(b1,b3,b4,b2)
   iff
      b5 in b1 /\ b2 &
       (for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
       for b7 being Element of REAL
          st b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) & b6 . 0 = b3 & b6 . 1 = b4 & b6 . b7 = b5 & 0 <= b7 & b7 <= 1
       for b8 being Element of REAL
             st b8 <= 1 & b7 < b8
          holds not b6 . b8 in b2);

:: JORDAN5C:th 5
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b3 in b1 & b1 is_an_arc_of b4,b5 & b2 = {b3}
   holds Last_Point(b1,b4,b5,b2) = b3;

:: JORDAN5C:th 6
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b4 in b2 & b1 /\ b2 is closed(TOP-REAL 2) & b1 is_an_arc_of b3,b4
   holds Last_Point(b1,b3,b4,b2) = b4;

:: JORDAN5C:th 7
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 c= b2 & b1 is closed(TOP-REAL 2) & b1 is_an_arc_of b3,b4
   holds First_Point(b1,b3,b4,b2) = b3 & Last_Point(b1,b3,b4,b2) = b4;

:: JORDAN5C:prednot 1 => JORDAN5C:pred 1
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  let a2, a3, a4, a5 be Element of the carrier of TOP-REAL 2;
  pred LE A4,A5,A1,A2,A3 means
    a4 in a1 &
     a5 in a1 &
     (for b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | a1
     for b2, b3 being Element of REAL
           st b1 is being_homeomorphism(I[01], (TOP-REAL 2) | a1) & b1 . 0 = a2 & b1 . 1 = a3 & b1 . b2 = a4 & 0 <= b2 & b2 <= 1 & b1 . b3 = a5 & 0 <= b3 & b3 <= 1
        holds b2 <= b3);
end;

:: JORDAN5C:dfs 3
definiens
  let a1 be Element of bool the carrier of TOP-REAL 2;
  let a2, a3, a4, a5 be Element of the carrier of TOP-REAL 2;
To prove
     LE a4,a5,a1,a2,a3
it is sufficient to prove
  thus a4 in a1 &
     a5 in a1 &
     (for b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | a1
     for b2, b3 being Element of REAL
           st b1 is being_homeomorphism(I[01], (TOP-REAL 2) | a1) & b1 . 0 = a2 & b1 . 1 = a3 & b1 . b2 = a4 & 0 <= b2 & b2 <= 1 & b1 . b3 = a5 & 0 <= b3 & b3 <= 1
        holds b2 <= b3);

:: JORDAN5C:def 3
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2 holds
   LE b4,b5,b1,b2,b3
iff
   b4 in b1 &
    b5 in b1 &
    (for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
    for b7, b8 being Element of REAL
          st b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) & b6 . 0 = b2 & b6 . 1 = b3 & b6 . b7 = b4 & 0 <= b7 & b7 <= 1 & b6 . b8 = b5 & 0 <= b8 & b8 <= 1
       holds b7 <= b8);

:: JORDAN5C:th 8
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b7, b8 being Element of REAL
      st b1 is_an_arc_of b2,b3 & b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) & b6 . 0 = b2 & b6 . 1 = b3 & b6 . b7 = b4 & 0 <= b7 & b7 <= 1 & b6 . b8 = b5 & 0 <= b8 & b8 <= 1 & b7 <= b8
   holds LE b4,b5,b1,b2,b3;

:: JORDAN5C:th 9
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b4 in b1
   holds LE b4,b4,b1,b2,b3;

:: JORDAN5C:th 10
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3 & b4 in b1
   holds LE b2,b4,b1,b2,b3 & LE b4,b3,b1,b2,b3;

:: JORDAN5C:th 11
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3
   holds LE b2,b3,b1,b2,b3;

:: JORDAN5C:th 12
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3 & LE b4,b5,b1,b2,b3 & LE b5,b4,b1,b2,b3
   holds b4 = b5;

:: JORDAN5C:th 13
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5, b6 being Element of the carrier of TOP-REAL 2
      st LE b4,b5,b1,b2,b3 & LE b5,b6,b1,b2,b3
   holds LE b4,b6,b1,b2,b3;

:: JORDAN5C:th 14
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b2,b3 & b4 in b1 & b5 in b1 & b4 <> b5 & (LE b4,b5,b1,b2,b3 implies LE b5,b4,b1,b2,b3)
   holds LE b5,b4,b1,b2,b3 & not LE b4,b5,b1,b2,b3;

:: JORDAN5C:th 15
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
      st b1 is being_S-Seq & (L~ b1) /\ b2 is closed(TOP-REAL 2) & b3 in L~ b1 & b3 in b2
   holds LE First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2),b3,L~ b1,b1 /. 1,b1 /. len b1;

:: JORDAN5C:th 16
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
      st b1 is being_S-Seq & (L~ b1) /\ b2 is closed(TOP-REAL 2) & b3 in L~ b1 & b3 in b2
   holds LE b3,Last_Point(L~ b1,b1 /. 1,b1 /. len b1,b2),L~ b1,b1 /. 1,b1 /. len b1;

:: JORDAN5C:th 17
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st b3 <> b4 & LE b1,b2,LSeg(b3,b4),b3,b4
   holds LE b1,b2,b3,b4;

:: JORDAN5C:th 18
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b1 is_an_arc_of b3,b4 & b1 meets b2 & b1 /\ b2 is closed(TOP-REAL 2)
   holds First_Point(b1,b3,b4,b2) = Last_Point(b1,b4,b3,b2) & Last_Point(b1,b3,b4,b2) = First_Point(b1,b4,b3,b2);

:: JORDAN5C:th 19
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of NAT
      st L~ b1 meets b2 &
         b2 is closed(TOP-REAL 2) &
         b1 is being_S-Seq &
         1 <= b3 &
         b3 + 1 <= len b1 &
         First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) in LSeg(b1,b3)
   holds First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) = First_Point(LSeg(b1,b3),b1 /. b3,b1 /. (b3 + 1),b2);

:: JORDAN5C:th 20
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of NAT
      st L~ b1 meets b2 &
         b2 is closed(TOP-REAL 2) &
         b1 is being_S-Seq &
         1 <= b3 &
         b3 + 1 <= len b1 &
         Last_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) in LSeg(b1,b3)
   holds Last_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) = Last_Point(LSeg(b1,b3),b1 /. b3,b1 /. (b3 + 1),b2);

:: JORDAN5C:th 21
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 1 <= b2 &
         b2 + 1 <= len b1 &
         b1 is being_S-Seq &
         First_Point(L~ b1,b1 /. 1,b1 /. len b1,LSeg(b1,b2)) in LSeg(b1,b2)
   holds First_Point(L~ b1,b1 /. 1,b1 /. len b1,LSeg(b1,b2)) = b1 /. b2;

:: JORDAN5C:th 22
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 1 <= b2 &
         b2 + 1 <= len b1 &
         b1 is being_S-Seq &
         Last_Point(L~ b1,b1 /. 1,b1 /. len b1,LSeg(b1,b2)) in LSeg(b1,b2)
   holds Last_Point(L~ b1,b1 /. 1,b1 /. len b1,LSeg(b1,b2)) = b1 /. (b2 + 1);

:: JORDAN5C:th 23
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st b1 is being_S-Seq & 1 <= b2 & b2 + 1 <= len b1
   holds LE b1 /. b2,b1 /. (b2 + 1),L~ b1,b1 /. 1,b1 /. len b1;

:: JORDAN5C:th 24
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 <= b3 & b3 <= len b1
   holds LE b1 /. b2,b1 /. b3,L~ b1,b1 /. 1,b1 /. len b1;

:: JORDAN5C:th 25
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 being_S-Seq & 1 <= b3 & b3 + 1 <= len b1 & b2 in LSeg(b1,b3)
   holds LE b1 /. b3,b2,L~ b1,b1 /. 1,b1 /. len b1;

:: JORDAN5C:th 26
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 being_S-Seq & 1 <= b3 & b3 + 1 <= len b1 & b2 in LSeg(b1,b3)
   holds LE b2,b1 /. (b3 + 1),L~ b1,b1 /. 1,b1 /. len b1;

:: JORDAN5C:th 27
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
for b4, b5 being Element of NAT
      st L~ b1 meets b2 &
         b1 is being_S-Seq &
         b2 is closed(TOP-REAL 2) &
         First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) in LSeg(b1,b4) &
         1 <= b4 &
         b4 + 1 <= len b1 &
         b3 in LSeg(b1,b5) &
         1 <= b5 &
         b5 + 1 <= len b1 &
         b3 in b2 &
         First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) <> b3
   holds b4 <= b5 &
    (b4 = b5 implies LE First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2),b3,b1 /. b4,b1 /. (b4 + 1));

:: JORDAN5C:th 28
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
for b4, b5 being Element of NAT
      st L~ b1 meets b2 &
         b1 is being_S-Seq &
         b2 is closed(TOP-REAL 2) &
         Last_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) in LSeg(b1,b4) &
         1 <= b4 &
         b4 + 1 <= len b1 &
         b3 in LSeg(b1,b5) &
         1 <= b5 &
         b5 + 1 <= len b1 &
         b3 in b2 &
         Last_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) <> b3
   holds b5 <= b4 &
    (b4 = b5 implies LE b3,Last_Point(L~ b1,b1 /. 1,b1 /. len b1,b2),b1 /. b4,b1 /. (b4 + 1));

:: JORDAN5C:th 29
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being Element of NAT
      st b2 in LSeg(b1,b4) &
         b3 in LSeg(b1,b4) &
         b1 is being_S-Seq &
         1 <= b4 &
         b4 + 1 <= len b1 &
         LE b2,b3,L~ b1,b1 /. 1,b1 /. len b1
   holds LE b2,b3,LSeg(b1,b4),b1 /. b4,b1 /. (b4 + 1);

:: JORDAN5C:th 30
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 & b1 is being_S-Seq & b2 <> b3
   holds    LE b2,b3,L~ b1,b1 /. 1,b1 /. len b1
   iff
      for b4, b5 being Element of NAT
            st b2 in LSeg(b1,b4) & b3 in LSeg(b1,b5) & 1 <= b4 & b4 + 1 <= len b1 & 1 <= b5 & b5 + 1 <= len b1
         holds b4 <= b5 &
          (b4 = b5 implies LE b2,b3,b1 /. b4,b1 /. (b4 + 1));