Article JORDAN7, MML version 4.99.1005

:: JORDAN7:th 1
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve
   holds W-min b1 in Lower_Arc b1 & E-max b1 in Lower_Arc b1 & W-min b1 in Upper_Arc b1 & E-max b1 in Upper_Arc b1;

:: JORDAN7:th 2
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve & LE b2,W-min b1,b1
   holds b2 = W-min b1;

:: JORDAN7:th 3
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve & b2 in b1
   holds LE W-min b1,b2,b1;

:: JORDAN7:funcnot 1 => JORDAN7:func 1
definition
  let a1 be non empty compact Element of bool the carrier of TOP-REAL 2;
  let a2, a3 be Element of the carrier of TOP-REAL 2;
  func Segment(A2,A3,A1) -> Element of bool the carrier of TOP-REAL 2 equals
    {b1 where b1 is Element of the carrier of TOP-REAL 2: LE a2,b1,a1 & LE b1,a3,a1}
    if a3 <> W-min a1
    otherwise {b1 where b1 is Element of the carrier of TOP-REAL 2: (LE a2,b1,a1 or a2 in a1 & b1 = W-min a1)};
end;

:: JORDAN7:def 1
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2 holds
(b3 = W-min b1 or Segment(b2,b3,b1) = {b4 where b4 is Element of the carrier of TOP-REAL 2: LE b2,b4,b1 & LE b4,b3,b1}) &
 (b3 = W-min b1 implies Segment(b2,b3,b1) = {b4 where b4 is Element of the carrier of TOP-REAL 2: (LE b2,b4,b1 or b2 in b1 & b4 = W-min b1)});

:: JORDAN7:th 4
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve
   holds Segment(W-min b1,E-max b1,b1) = Upper_Arc b1 &
    Segment(E-max b1,W-min b1,b1) = Lower_Arc b1;

:: JORDAN7:th 5
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve & LE b2,b3,b1
   holds b2 in b1 & b3 in b1;

:: JORDAN7:th 6
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve & LE b2,b3,b1
   holds b2 in Segment(b2,b3,b1) & b3 in Segment(b2,b3,b1);

:: JORDAN7:th 7
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in b1 & b1 is being_simple_closed_curve
   holds b2 in Segment(b2,W-min b1,b1);

:: JORDAN7:th 8
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve & b2 in b1 & b2 <> W-min b1
   holds Segment(b2,b2,b1) = {b2};

:: JORDAN7:th 9
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve & b2 <> W-min b1 & b3 <> W-min b1
   holds not W-min b1 in Segment(b2,b3,b1);

:: JORDAN7:th 10
theorem
for b1 being non empty compact 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 being_simple_closed_curve & LE b2,b3,b1 & LE b3,b4,b1 & (b2 = b3 implies b2 <> W-min b1) & b2 <> b4 & (b3 = b4 implies b3 <> W-min b1)
   holds (Segment(b2,b3,b1)) /\ Segment(b3,b4,b1) = {b3};

:: JORDAN7:th 11
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve & LE b2,b3,b1 & b2 <> W-min b1 & b3 <> W-min b1
   holds (Segment(b2,b3,b1)) /\ Segment(b3,W-min b1,b1) = {b3};

:: JORDAN7:th 12
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 is being_simple_closed_curve & LE b2,b3,b1 & b2 <> b3 & b2 <> W-min b1
   holds (Segment(b3,W-min b1,b1)) /\ Segment(W-min b1,b2,b1) = {W-min b1};

:: JORDAN7:th 13
theorem
for b1 being non empty compact 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 being_simple_closed_curve & LE b2,b3,b1 & LE b3,b4,b1 & LE b4,b5,b1 & b2 <> b3 & b3 <> b4
   holds Segment(b2,b3,b1) misses Segment(b4,b5,b1);

:: JORDAN7:th 14
theorem
for b1 being non empty compact 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 being_simple_closed_curve & LE b2,b3,b1 & LE b3,b4,b1 & b2 <> W-min b1 & b2 <> b3 & b3 <> b4
   holds Segment(b2,b3,b1) misses Segment(b4,W-min b1,b1);

:: JORDAN7:th 15
theorem
for b1 being Element of NAT
for b2 being non empty Element of bool the carrier of TOP-REAL b1
for b3 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL b1) | b2
      st b2 <> {} & b3 is being_homeomorphism(I[01], (TOP-REAL b1) | b2)
   holds ex b4 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL b1 st
      b3 = b4 & b4 is continuous(I[01], TOP-REAL b1) & b4 is one-to-one;

:: JORDAN7:th 16
theorem
for b1 being Element of NAT
for b2 being non empty Element of bool the carrier of TOP-REAL b1
for b3 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL b1
      st b3 is continuous(I[01], TOP-REAL b1) & b3 is one-to-one & rng b3 = b2
   holds ex b4 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL b1) | b2 st
      b4 = b3 & b4 is being_homeomorphism(I[01], (TOP-REAL b1) | b2);

:: JORDAN7:condreg 1
registration
  cluster increasing -> one-to-one (FinSequence of REAL);
end;

:: JORDAN7:th 17
theorem
for b1 being FinSequence of REAL
      st b1 is increasing
   holds b1 is one-to-one;

:: JORDAN7:th 18
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 ex b4 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2 st
      b4 is continuous(I[01], TOP-REAL 2) & b4 is one-to-one & rng b4 = b1 & b4 . 0 = b2 & b4 . 1 = b3;

:: JORDAN7:th 19
theorem
for b1 being non empty 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
for b7, b8 being Element of REAL
      st b1 is_an_arc_of b2,b3 & b6 is continuous(I[01], TOP-REAL 2) & b6 is one-to-one & rng b6 = 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;

:: JORDAN7:th 20
theorem
for b1 being non empty 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
for b7, b8 being Element of REAL
      st b6 is continuous(I[01], TOP-REAL 2) & b6 is one-to-one & rng b6 = b1 & b6 . 0 = b2 & b6 . 1 = b3 & b6 . b7 = b4 & 0 <= b7 & b7 <= 1 & b6 . b8 = b5 & 0 <= b8 & b8 <= 1 & LE b4,b5,b1,b2,b3
   holds b7 <= b8;

:: JORDAN7:th 21
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
for b2 being Element of REAL
      st b1 is being_simple_closed_curve & 0 < b2
   holds ex b3 being FinSequence of the carrier of TOP-REAL 2 st
      b3 . 1 = W-min b1 &
       b3 is one-to-one &
       8 <= len b3 &
       rng b3 c= b1 &
       (for b4 being Element of NAT
             st 1 <= b4 & b4 < len b3
          holds LE b3 /. b4,b3 /. (b4 + 1),b1) &
       (for b4 being Element of NAT
       for b5 being Element of bool the carrier of Euclid 2
             st 1 <= b4 &
                b4 < len b3 &
                b5 = Segment(b3 /. b4,b3 /. (b4 + 1),b1)
          holds diameter b5 < b2) &
       (for b4 being Element of bool the carrier of Euclid 2
             st b4 = Segment(b3 /. len b3,b3 /. 1,b1)
          holds diameter b4 < b2) &
       (for b4 being Element of NAT
             st 1 <= b4 & b4 + 1 < len b3
          holds (Segment(b3 /. b4,b3 /. (b4 + 1),b1)) /\ Segment(b3 /. (b4 + 1),b3 /. (b4 + 2),b1) = {b3 /. (b4 + 1)}) &
       (Segment(b3 /. len b3,b3 /. 1,b1)) /\ Segment(b3 /. 1,b3 /. 2,b1) = {b3 /. 1} &
       (Segment(b3 /. ((len b3) -' 1),b3 /. len b3,b1)) /\ Segment(b3 /. len b3,b3 /. 1,b1) = {b3 /. len b3} &
       Segment(b3 /. ((len b3) -' 1),b3 /. len b3,b1) misses Segment(b3 /. 1,b3 /. 2,b1) &
       (for b4, b5 being Element of NAT
             st 1 <= b4 & b4 < b5 & b5 < len b3 & not b4,b5 are_adjacent1
          holds Segment(b3 /. b4,b3 /. (b4 + 1),b1) misses Segment(b3 /. b5,b3 /. (b5 + 1),b1)) &
       (for b4 being Element of NAT
             st 1 < b4 & b4 + 1 < len b3
          holds Segment(b3 /. len b3,b3 /. 1,b1) misses Segment(b3 /. b4,b3 /. (b4 + 1),b1));