Article JORDAN17, MML version 4.99.1005

:: JORDAN17:th 1
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1
for b5 being Element of bool the carrier of TOP-REAL b1
      st b2 in b5 & b5 is_an_arc_of b3,b4
   holds ex b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL b1) | b5 st
      ex b7 being Element of REAL st
         b6 is being_homeomorphism(I[01], (TOP-REAL b1) | b5) & b6 . 0 = b3 & b6 . 1 = b4 & 0 <= b7 & b7 <= 1 & b6 . b7 = b2;

:: JORDAN17:th 2
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   LE W-min b1,E-max b1,b1;

:: JORDAN17:th 3
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st LE b2,E-max b1,b1
   holds b2 in Upper_Arc b1;

:: JORDAN17:th 4
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st LE E-max b1,b2,b1
   holds b2 in Lower_Arc b1;

:: JORDAN17:th 5
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st LE b2,W-min b1,b1
   holds b2 in Lower_Arc b1;

:: JORDAN17:th 6
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being Element of bool the carrier of TOP-REAL 2
      st b1 <> b2 & b5 is_an_arc_of b3,b4 & LE b1,b2,b5,b3,b4
   holds ex b6 being Element of the carrier of TOP-REAL 2 st
      b1 <> b6 & b2 <> b6 & LE b1,b6,b5,b3,b4 & LE b6,b2,b5,b3,b4;

:: JORDAN17:th 7
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in b1
   holds ex b3 being Element of the carrier of TOP-REAL 2 st
      b2 <> b3 & LE b2,b3,b1;

:: JORDAN17:th 8
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & LE b2,b3,b1
   holds ex b4 being Element of the carrier of TOP-REAL 2 st
      b4 <> b2 & b4 <> b3 & LE b2,b4,b1 & LE b4,b3,b1;

:: JORDAN17:prednot 1 => JORDAN17: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 A2,A3,A4,A5 are_in_this_order_on A1 means
    ((LE a2,a3,a1 & LE a3,a4,a1 implies not LE a4,a5,a1) &
     (LE a3,a4,a1 & LE a4,a5,a1 implies not LE a5,a2,a1) &
     (LE a4,a5,a1 & LE a5,a2,a1 implies not LE a2,a3,a1)) implies LE a5,a2,a1 & LE a2,a3,a1 & LE a3,a4,a1;
end;

:: JORDAN17:dfs 1
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
     a2,a3,a4,a5 are_in_this_order_on a1
it is sufficient to prove
  thus ((LE a2,a3,a1 & LE a3,a4,a1 implies not LE a4,a5,a1) &
     (LE a3,a4,a1 & LE a4,a5,a1 implies not LE a5,a2,a1) &
     (LE a4,a5,a1 & LE a5,a2,a1 implies not LE a2,a3,a1)) implies LE a5,a2,a1 & LE a2,a3,a1 & LE a3,a4,a1;

:: JORDAN17:def 1
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
   b2,b3,b4,b5 are_in_this_order_on b1
iff
   ((LE b2,b3,b1 & LE b3,b4,b1 implies not LE b4,b5,b1) &
    (LE b3,b4,b1 & LE b4,b5,b1 implies not LE b5,b2,b1) &
    (LE b4,b5,b1 & LE b5,b2,b1 implies not LE b2,b3,b1) implies LE b5,b2,b1 & LE b2,b3,b1 & LE b3,b4,b1);

:: JORDAN17:th 9
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in b1
   holds b2,b2,b2,b2 are_in_this_order_on b1;

:: JORDAN17:th 10
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2,b3,b4,b5 are_in_this_order_on b1
   holds b3,b4,b5,b2 are_in_this_order_on b1;

:: JORDAN17:th 11
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2,b3,b4,b5 are_in_this_order_on b1
   holds b4,b5,b2,b3 are_in_this_order_on b1;

:: JORDAN17:th 12
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2,b3,b4,b5 are_in_this_order_on b1
   holds b5,b2,b3,b4 are_in_this_order_on b1;

:: JORDAN17:th 13
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b2,b3,b4,b5 are_in_this_order_on b1
   holds ex b6 being Element of the carrier of TOP-REAL 2 st
      b6 <> b2 & b6 <> b3 & b2,b6,b3,b4 are_in_this_order_on b1;

:: JORDAN17:th 14
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b2,b3,b4,b5 are_in_this_order_on b1
   holds ex b6 being Element of the carrier of TOP-REAL 2 st
      b6 <> b2 & b6 <> b3 & b2,b6,b3,b5 are_in_this_order_on b1;

:: JORDAN17:th 15
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b4,b2,b3,b5 are_in_this_order_on b1
   holds ex b6 being Element of the carrier of TOP-REAL 2 st
      b6 <> b2 & b6 <> b3 & b4,b2,b6,b3 are_in_this_order_on b1;

:: JORDAN17:th 16
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b4,b2,b3,b5 are_in_this_order_on b1
   holds ex b6 being Element of the carrier of TOP-REAL 2 st
      b6 <> b2 & b6 <> b3 & b2,b6,b3,b5 are_in_this_order_on b1;

:: JORDAN17:th 17
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b4,b5,b2,b3 are_in_this_order_on b1
   holds ex b6 being Element of the carrier of TOP-REAL 2 st
      b6 <> b2 & b6 <> b3 & b4,b2,b6,b3 are_in_this_order_on b1;

:: JORDAN17:th 18
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b4,b5,b2,b3 are_in_this_order_on b1
   holds ex b6 being Element of the carrier of TOP-REAL 2 st
      b6 <> b2 & b6 <> b3 & b5,b2,b6,b3 are_in_this_order_on b1;

:: JORDAN17:th 19
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b3,b4,b5,b2 are_in_this_order_on b1
   holds ex b6 being Element of the carrier of TOP-REAL 2 st
      b6 <> b2 & b6 <> b3 & b3,b4,b2,b6 are_in_this_order_on b1;

:: JORDAN17:th 20
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b3,b4,b5,b2 are_in_this_order_on b1
   holds ex b6 being Element of the carrier of TOP-REAL 2 st
      b6 <> b2 & b6 <> b3 & b3,b5,b2,b6 are_in_this_order_on b1;

:: JORDAN17:th 21
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b2 <> b4 & b5 <> b4 & b2,b5,b3,b4 are_in_this_order_on b1 & b5,b2,b3,b4 are_in_this_order_on b1
   holds b2 = b5;

:: JORDAN17:th 22
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b3 <> b4 & b4 <> b5 & b2,b3,b4,b5 are_in_this_order_on b1 & b4,b3,b2,b5 are_in_this_order_on b1
   holds b2 = b4;

:: JORDAN17:th 23
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b2 <> b4 & b3 <> b5 & b2,b3,b4,b5 are_in_this_order_on b1 & b5,b3,b4,b2 are_in_this_order_on b1
   holds b2 = b5;

:: JORDAN17:th 24
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b2 <> b4 & b5 <> b4 & b2,b5,b3,b4 are_in_this_order_on b1 & b2,b3,b5,b4 are_in_this_order_on b1
   holds b5 = b3;

:: JORDAN17:th 25
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b3 <> b4 & b4 <> b5 & b2,b3,b4,b5 are_in_this_order_on b1 & b2,b5,b4,b3 are_in_this_order_on b1
   holds b3 = b5;

:: JORDAN17:th 26
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b2 <> b4 & b3 <> b5 & b2,b3,b4,b5 are_in_this_order_on b1 & b2,b3,b5,b4 are_in_this_order_on b1
   holds b4 = b5;

:: JORDAN17:th 27
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b2 in b1 & b3 in b1 & b4 in b1 & b5 in b1 & not b2,b3,b4,b5 are_in_this_order_on b1 & not b2,b3,b5,b4 are_in_this_order_on b1 & not b2,b4,b3,b5 are_in_this_order_on b1 & not b2,b4,b5,b3 are_in_this_order_on b1 & not b2,b5,b3,b4 are_in_this_order_on b1
   holds b2,b5,b4,b3 are_in_this_order_on b1;