Article JORDAN21, MML version 4.99.1005

:: JORDAN21:th 1
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1 holds
   {b2} is Bounded(b1);

:: JORDAN21:th 2
theorem
for b1, b2 being real set
for b3 being Element of bool the carrier of TOP-REAL 2
      st b3 = {|[b4,b2]| where b4 is Element of REAL: b1 < b4}
   holds b3 is convex(2);

:: JORDAN21:th 3
theorem
for b1, b2 being real set
for b3 being Element of bool the carrier of TOP-REAL 2
      st b3 = {|[b4,b2]| where b4 is Element of REAL: b4 < b1}
   holds b3 is convex(2);

:: JORDAN21:th 4
theorem
for b1, b2 being real set
for b3 being Element of bool the carrier of TOP-REAL 2
      st b3 = {|[b1,b4]| where b4 is Element of REAL: b2 < b4}
   holds b3 is convex(2);

:: JORDAN21:th 5
theorem
for b1, b2 being real set
for b3 being Element of bool the carrier of TOP-REAL 2
      st b3 = {|[b1,b4]| where b4 is Element of REAL: b4 < b2}
   holds b3 is convex(2);

:: JORDAN21:th 6
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   (north_halfline b1) \ {b1} is convex(2);

:: JORDAN21:th 7
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   (south_halfline b1) \ {b1} is convex(2);

:: JORDAN21:th 8
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   (west_halfline b1) \ {b1} is convex(2);

:: JORDAN21:th 9
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   (east_halfline b1) \ {b1} is convex(2);

:: JORDAN21:th 10
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   UBD b1 misses b1;

:: JORDAN21:th 11
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 & b2 <> b4 & b3 <> b5
   holds not b2 in Segment(b1,b2,b3,b4,b5) & not b3 in Segment(b1,b2,b3,b4,b5);

:: JORDAN21:attrnot 1 => JORDAN21:attr 1
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  attr a1 is with_the_max_arc means
    a1 meets Vertical_Line (((W-bound a1) + E-bound a1) / 2);
end;

:: JORDAN21:dfs 1
definiens
  let a1 be Element of bool the carrier of TOP-REAL 2;
To prove
     a1 is with_the_max_arc
it is sufficient to prove
  thus a1 meets Vertical_Line (((W-bound a1) + E-bound a1) / 2);

:: JORDAN21:def 1
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
      b1 is with_the_max_arc
   iff
      b1 meets Vertical_Line (((W-bound b1) + E-bound b1) / 2);

:: JORDAN21:condreg 1
registration
  cluster with_the_max_arc -> non empty (Element of bool the carrier of TOP-REAL 2);
end;

:: JORDAN21:condreg 2
registration
  cluster being_simple_closed_curve -> with_the_max_arc (Element of bool the carrier of TOP-REAL 2);
end;

:: JORDAN21:exreg 1
registration
  cluster non empty closed connected compact non horizontal non vertical being_simple_closed_curve with_the_max_arc Element of bool the carrier of TOP-REAL 2;
end;

:: JORDAN21:th 12
theorem
for b1 being with_the_max_arc Element of bool the carrier of TOP-REAL 2 holds
   proj2 .: (b1 /\ Vertical_Line (((W-bound b1) + E-bound b1) / 2)) is not empty;

:: JORDAN21:th 13
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2 holds
   proj2 .: (b1 /\ Vertical_Line (((W-bound b1) + E-bound b1) / 2)) is closed &
    proj2 .: (b1 /\ Vertical_Line (((W-bound b1) + E-bound b1) / 2)) is bounded_below &
    proj2 .: (b1 /\ Vertical_Line (((W-bound b1) + E-bound b1) / 2)) is bounded_above;

:: JORDAN21:th 23
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   Lower_Middle_Point b1 in b1;

:: JORDAN21:th 24
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   (Lower_Middle_Point b1) `2 <> (Upper_Middle_Point b1) `2;

:: JORDAN21:th 25
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   Lower_Middle_Point b1 <> Upper_Middle_Point b1;

:: JORDAN21:th 26
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   W-bound b1 = W-bound Upper_Arc b1;

:: JORDAN21:th 27
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   E-bound b1 = E-bound Upper_Arc b1;

:: JORDAN21:th 28
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   W-bound b1 = W-bound Lower_Arc b1;

:: JORDAN21:th 29
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   E-bound b1 = E-bound Lower_Arc b1;

:: JORDAN21:th 30
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   (Upper_Arc b1) /\ Vertical_Line (((W-bound b1) + E-bound b1) / 2) is not empty &
    proj2 .: ((Upper_Arc b1) /\ Vertical_Line (((W-bound b1) + E-bound b1) / 2)) is not empty;

:: JORDAN21:th 31
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   (Lower_Arc b1) /\ Vertical_Line (((W-bound b1) + E-bound b1) / 2) is not empty &
    proj2 .: ((Lower_Arc b1) /\ Vertical_Line (((W-bound b1) + E-bound b1) / 2)) is not empty;

:: JORDAN21:th 32
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being connected compact Element of bool the carrier of TOP-REAL 2
      st b2 c= b1 & W-min b1 in b2 & E-max b1 in b2 & not Upper_Arc b1 c= b2
   holds Lower_Arc b1 c= b2;

:: JORDAN21:funcreg 1
registration
  let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
  cluster Lower_Arc a1 -> non empty with_the_max_arc;
end;

:: JORDAN21:funcreg 2
registration
  let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
  cluster Upper_Arc a1 -> non empty with_the_max_arc;
end;

:: JORDAN21:funcnot 1 => JORDAN21:func 1
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func UMP A1 -> Element of the carrier of TOP-REAL 2 equals
    |[((E-bound a1) + W-bound a1) / 2,sup (proj2 .: (a1 /\ Vertical_Line (((E-bound a1) + W-bound a1) / 2)))]|;
end;

:: JORDAN21:def 2
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   UMP b1 = |[((E-bound b1) + W-bound b1) / 2,sup (proj2 .: (b1 /\ Vertical_Line (((E-bound b1) + W-bound b1) / 2)))]|;

:: JORDAN21:funcnot 2 => JORDAN21:func 2
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  func LMP A1 -> Element of the carrier of TOP-REAL 2 equals
    |[((E-bound a1) + W-bound a1) / 2,inf (proj2 .: (a1 /\ Vertical_Line (((E-bound a1) + W-bound a1) / 2)))]|;
end;

:: JORDAN21:def 3
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   LMP b1 = |[((E-bound b1) + W-bound b1) / 2,inf (proj2 .: (b1 /\ Vertical_Line (((E-bound b1) + W-bound b1) / 2)))]|;

:: JORDAN21:th 37
theorem
for b1 being compact non vertical Element of bool the carrier of TOP-REAL 2 holds
   UMP b1 <> W-min b1;

:: JORDAN21:th 38
theorem
for b1 being compact non vertical Element of bool the carrier of TOP-REAL 2 holds
   UMP b1 <> E-max b1;

:: JORDAN21:th 39
theorem
for b1 being compact non vertical Element of bool the carrier of TOP-REAL 2 holds
   LMP b1 <> W-min b1;

:: JORDAN21:th 40
theorem
for b1 being compact non vertical Element of bool the carrier of TOP-REAL 2 holds
   LMP b1 <> E-max b1;

:: JORDAN21:th 41
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 b2 /\ Vertical_Line (((W-bound b2) + E-bound b2) / 2)
   holds b1 `2 <= (UMP b2) `2;

:: JORDAN21:th 42
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 b2 /\ Vertical_Line (((W-bound b2) + E-bound b2) / 2)
   holds (LMP b2) `2 <= b1 `2;

:: JORDAN21:th 43
theorem
for b1 being compact with_the_max_arc Element of bool the carrier of TOP-REAL 2 holds
   UMP b1 in b1;

:: JORDAN21:th 44
theorem
for b1 being compact with_the_max_arc Element of bool the carrier of TOP-REAL 2 holds
   LMP b1 in b1;

:: JORDAN21:th 45
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   LSeg(UMP b1,|[((W-bound b1) + E-bound b1) / 2,N-bound b1]|) is vertical;

:: JORDAN21:th 46
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
   LSeg(LMP b1,|[((W-bound b1) + E-bound b1) / 2,S-bound b1]|) is vertical;

:: JORDAN21:th 47
theorem
for b1 being compact with_the_max_arc Element of bool the carrier of TOP-REAL 2 holds
   (LSeg(UMP b1,|[((W-bound b1) + E-bound b1) / 2,N-bound b1]|)) /\ b1 = {UMP b1};

:: JORDAN21:th 48
theorem
for b1 being compact with_the_max_arc Element of bool the carrier of TOP-REAL 2 holds
   (LSeg(LMP b1,|[((W-bound b1) + E-bound b1) / 2,S-bound b1]|)) /\ b1 = {LMP b1};

:: JORDAN21:th 49
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   (LMP b1) `2 < (UMP b1) `2;

:: JORDAN21:th 50
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   UMP b1 <> LMP b1;

:: JORDAN21:th 51
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   S-bound b1 < (UMP b1) `2;

:: JORDAN21:th 52
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   (UMP b1) `2 <= N-bound b1;

:: JORDAN21:th 53
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   S-bound b1 <= (LMP b1) `2;

:: JORDAN21:th 54
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   (LMP b1) `2 < N-bound b1;

:: JORDAN21:th 55
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   LSeg(UMP b1,|[((W-bound b1) + E-bound b1) / 2,N-bound b1]|) misses LSeg(LMP b1,|[((W-bound b1) + E-bound b1) / 2,S-bound b1]|);

:: JORDAN21:th 56
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 &
         (W-bound b1) + E-bound b1 = (W-bound b2) + E-bound b2 &
         b1 /\ Vertical_Line (((W-bound b1) + E-bound b1) / 2) is not empty &
         proj2 .: (b2 /\ Vertical_Line (((W-bound b1) + E-bound b1) / 2)) is bounded_above
   holds (UMP b1) `2 <= (UMP b2) `2;

:: JORDAN21:th 57
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 &
         (W-bound b1) + E-bound b1 = (W-bound b2) + E-bound b2 &
         b1 /\ Vertical_Line (((W-bound b1) + E-bound b1) / 2) is not empty &
         proj2 .: (b2 /\ Vertical_Line (((W-bound b1) + E-bound b1) / 2)) is bounded_below
   holds (LMP b2) `2 <= (LMP b1) `2;

:: JORDAN21:th 58
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 &
         UMP b2 in b1 &
         b1 /\ Vertical_Line (((W-bound b1) + E-bound b1) / 2) is not empty &
         proj2 .: (b2 /\ Vertical_Line (((W-bound b2) + E-bound b2) / 2)) is bounded_above &
         (W-bound b1) + E-bound b1 = (W-bound b2) + E-bound b2
   holds UMP b1 = UMP b2;

:: JORDAN21:th 59
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 &
         LMP b2 in b1 &
         b1 /\ Vertical_Line (((W-bound b1) + E-bound b1) / 2) is not empty &
         proj2 .: (b2 /\ Vertical_Line (((W-bound b2) + E-bound b2) / 2)) is bounded_below &
         (W-bound b1) + E-bound b1 = (W-bound b2) + E-bound b2
   holds LMP b1 = LMP b2;

:: JORDAN21:th 60
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   (UMP Upper_Arc b1) `2 <= N-bound b1;

:: JORDAN21:th 61
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   S-bound b1 <= (LMP Lower_Arc b1) `2;

:: JORDAN21:th 62
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
      st LMP b1 in Lower_Arc b1
   holds not UMP b1 in Lower_Arc b1;

:: JORDAN21:th 63
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
      st LMP b1 in Upper_Arc b1
   holds not UMP b1 in Upper_Arc b1;