Article JORDAN18, MML version 4.99.1005

:: JORDAN18:th 1
theorem
for b1, b2 being real set holds
(b1 - b2) ^2 = (b2 - b1) ^2;

:: JORDAN18:th 3
theorem
for b1, b2 being non empty TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b2
      st b3 is being_homeomorphism(b1, b2) & b4 is compact(b2)
   holds b3 " b4 is compact(b1);

:: JORDAN18:th 4
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   proj2 .: north_halfline b1 is bounded_below;

:: JORDAN18:th 5
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   proj2 .: south_halfline b1 is bounded_above;

:: JORDAN18:th 6
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   proj1 .: west_halfline b1 is bounded_above;

:: JORDAN18:th 7
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   proj1 .: east_halfline b1 is bounded_below;

:: JORDAN18:funcreg 1
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  cluster proj2 .: north_halfline a1 -> non empty;
end;

:: JORDAN18:funcreg 2
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  cluster proj2 .: south_halfline a1 -> non empty;
end;

:: JORDAN18:funcreg 3
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  cluster proj1 .: west_halfline a1 -> non empty;
end;

:: JORDAN18:funcreg 4
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  cluster proj1 .: east_halfline a1 -> non empty;
end;

:: JORDAN18:th 8
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   inf (proj2 .: north_halfline b1) = b1 `2;

:: JORDAN18:th 9
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   sup (proj2 .: south_halfline b1) = b1 `2;

:: JORDAN18:th 10
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   sup (proj1 .: west_halfline b1) = b1 `1;

:: JORDAN18:th 11
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   inf (proj1 .: east_halfline b1) = b1 `1;

:: JORDAN18:funcreg 5
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  cluster north_halfline a1 -> closed;
end;

:: JORDAN18:funcreg 6
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  cluster south_halfline a1 -> closed;
end;

:: JORDAN18:funcreg 7
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  cluster east_halfline a1 -> closed;
end;

:: JORDAN18:funcreg 8
registration
  let a1 be Element of the carrier of TOP-REAL 2;
  cluster west_halfline a1 -> closed;
end;

:: JORDAN18:th 12
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in BDD b1
   holds not north_halfline b2 c= UBD b1;

:: JORDAN18:th 13
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in BDD b1
   holds not south_halfline b2 c= UBD b1;

:: JORDAN18:th 14
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in BDD b1
   holds not east_halfline b2 c= UBD b1;

:: JORDAN18:th 15
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in BDD b1
   holds not west_halfline b2 c= UBD b1;

:: JORDAN18:th 18
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
      st b2 is_an_arc_of b3,b4 & b2 c= b1
   holds ex b5 being non empty Element of bool the carrier of TOP-REAL 2 st
      b5 is_an_arc_of b3,b4 & b2 \/ b5 = b1 & b2 /\ b5 = {b3,b4};

:: JORDAN18:funcnot 1 => JORDAN18:func 1
definition
  let a1 be Element of the carrier of TOP-REAL 2;
  let a2 be Element of bool the carrier of TOP-REAL 2;
  func North-Bound(A1,A2) -> Element of the carrier of TOP-REAL 2 equals
    |[a1 `1,inf (proj2 .: (a2 /\ north_halfline a1))]|;
end;

:: JORDAN18:def 1
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2 holds
   North-Bound(b1,b2) = |[b1 `1,inf (proj2 .: (b2 /\ north_halfline b1))]|;

:: JORDAN18:funcnot 2 => JORDAN18:func 2
definition
  let a1 be Element of the carrier of TOP-REAL 2;
  let a2 be Element of bool the carrier of TOP-REAL 2;
  func South-Bound(A1,A2) -> Element of the carrier of TOP-REAL 2 equals
    |[a1 `1,sup (proj2 .: (a2 /\ south_halfline a1))]|;
end;

:: JORDAN18:def 2
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2 holds
   South-Bound(b1,b2) = |[b1 `1,sup (proj2 .: (b2 /\ south_halfline b1))]|;

:: JORDAN18:th 20
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2 holds
   (North-Bound(b2,b1)) `1 = b2 `1 & (South-Bound(b2,b1)) `1 = b2 `1;

:: JORDAN18:th 22
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being compact Element of bool the carrier of TOP-REAL 2
      st b1 in BDD b2
   holds North-Bound(b1,b2) in b2 & North-Bound(b1,b2) in north_halfline b1 & South-Bound(b1,b2) in b2 & South-Bound(b1,b2) in south_halfline b1;

:: JORDAN18:th 23
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being compact Element of bool the carrier of TOP-REAL 2
      st b1 in BDD b2
   holds (South-Bound(b1,b2)) `2 < b1 `2 & b1 `2 < (North-Bound(b1,b2)) `2;

:: JORDAN18:th 24
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being compact Element of bool the carrier of TOP-REAL 2
      st b1 in BDD b2
   holds sup (proj2 .: (b2 /\ south_halfline b1)) < inf (proj2 .: (b2 /\ north_halfline b1));

:: JORDAN18:th 25
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being compact Element of bool the carrier of TOP-REAL 2
      st b1 in BDD b2
   holds South-Bound(b1,b2) <> North-Bound(b1,b2);

:: JORDAN18:th 26
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2 holds
   LSeg(North-Bound(b1,b2),South-Bound(b1,b2)) is vertical;

:: JORDAN18:th 27
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being compact Element of bool the carrier of TOP-REAL 2
      st b1 in BDD b2
   holds (LSeg(North-Bound(b1,b2),South-Bound(b1,b2))) /\ b2 = {North-Bound(b1,b2),South-Bound(b1,b2)};

:: JORDAN18:th 28
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being compact Element of bool the carrier of TOP-REAL 2
      st b1 in BDD b3 & b2 in BDD b3 & b1 `1 <> b2 `1
   holds North-Bound(b1,b3),South-Bound(b2,b3),North-Bound(b2,b3),South-Bound(b1,b3) are_mutually_different;

:: JORDAN18:prednot 1 => JORDAN18:pred 1
definition
  let a1 be Element of NAT;
  let a2 be Element of bool the carrier of TOP-REAL a1;
  let a3, a4, a5, a6 be Element of the carrier of TOP-REAL a1;
  pred A3,A4,A2 -separate A5,A6 means
    for b1 being Element of bool the carrier of TOP-REAL a1
          st b1 is_an_arc_of a3,a4 & b1 c= a2
       holds b1 meets {a5,a6};
end;

:: JORDAN18:dfs 3
definiens
  let a1 be Element of NAT;
  let a2 be Element of bool the carrier of TOP-REAL a1;
  let a3, a4, a5, a6 be Element of the carrier of TOP-REAL a1;
To prove
     a3,a4,a2 -separate a5,a6
it is sufficient to prove
  thus for b1 being Element of bool the carrier of TOP-REAL a1
          st b1 is_an_arc_of a3,a4 & b1 c= a2
       holds b1 meets {a5,a6};

:: JORDAN18:def 3
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4, b5, b6 being Element of the carrier of TOP-REAL b1 holds
   b3,b4,b2 -separate b5,b6
iff
   for b7 being Element of bool the carrier of TOP-REAL b1
         st b7 is_an_arc_of b3,b4 & b7 c= b2
      holds b7 meets {b5,b6};

:: JORDAN18:prednot 2 => not JORDAN18:pred 1
notation
  let a1 be Element of NAT;
  let a2 be Element of bool the carrier of TOP-REAL a1;
  let a3, a4, a5, a6 be Element of the carrier of TOP-REAL a1;
  antonym a3,a4 are_neighbours_wrt a5,a6,a2 for a3,a4,a2 -separate a5,a6;
end;

:: JORDAN18:th 29
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4, b5 being Element of the carrier of TOP-REAL b1 holds
b3,b3,b2 -separate b4,b5;

:: JORDAN18:th 30
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4, b5, b6 being Element of the carrier of TOP-REAL b1
      st b3,b4,b2 -separate b5,b6
   holds b4,b3,b2 -separate b5,b6;

:: JORDAN18:th 31
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4, b5, b6 being Element of the carrier of TOP-REAL b1
      st b3,b4,b2 -separate b5,b6
   holds b3,b4,b2 -separate b6,b5;

:: JORDAN18:th 32
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4, b5 being Element of the carrier of TOP-REAL b1 holds
b3,b4,b2 -separate b3,b5;

:: JORDAN18:th 33
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4, b5 being Element of the carrier of TOP-REAL b1 holds
b3,b4,b2 -separate b5,b4;

:: JORDAN18:th 34
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4, b5 being Element of the carrier of TOP-REAL b1 holds
b3,b4,b2 -separate b4,b5;

:: JORDAN18:th 35
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4, b5 being Element of the carrier of TOP-REAL b1 holds
b3,b4,b2 -separate b5,b3;

:: JORDAN18:th 36
theorem
for b1 being being_simple_closed_curve 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 & b2 in b1 & b3 in b1 & b2 <> b3 & b2 <> b4 & b3 <> b4
   holds b2,b3 are_neighbours_wrt b4,b4,b1;

:: JORDAN18:th 37
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 in b1 & b3 in b1 & b2,b3,b1 -separate b4,b5
   holds b4,b5,b1 -separate b2,b3;

:: JORDAN18:th 38
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 & b2 <> b3 & b4 <> b2 & b4 <> b3 & b5 <> b2 & b5 <> b3 & b2,b3,b1 -separate b4,b5
   holds b2,b4 are_neighbours_wrt b3,b5,b1;