Article SPPOL_2, MML version 4.99.1005

:: SPPOL_2:th 1
theorem
for b1, b2, b3, b4 being real set
      st |[b1,b2]| = |[b3,b4]|
   holds b1 = b3 & b2 = b4;

:: SPPOL_2:th 2
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being natural set
      st b2 + b3 = len b1
   holds LSeg(b1,b2) = LSeg(Rev b1,b3);

:: SPPOL_2:th 3
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being natural set
      st b2 + 1 <= len (b1 | b3)
   holds LSeg(b1 | b3,b2) = LSeg(b1,b2);

:: SPPOL_2:th 4
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being natural set
      st b3 <= len b1 & 1 <= b2
   holds LSeg(b1 /^ b3,b2) = LSeg(b1,b3 + b2);

:: SPPOL_2:th 5
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being natural set
      st 1 <= b2 & b2 + 1 <= (len b1) - b3
   holds LSeg(b1 /^ b3,b2) = LSeg(b1,b3 + b2);

:: SPPOL_2:th 6
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being natural set
      st b3 + 1 <= len b1
   holds LSeg(b1 ^ b2,b3) = LSeg(b1,b3);

:: SPPOL_2:th 7
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being natural set
      st 1 <= b3
   holds LSeg(b1 ^ b2,(len b1) + b3) = LSeg(b2,b3);

:: SPPOL_2:th 8
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
      st b1 is not empty & b2 is not empty
   holds LSeg(b1 ^ b2,len b1) = LSeg(b1 /. len b1,b2 /. 1);

:: SPPOL_2:th 9
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 natural set
      st b3 + 1 <= len (b1 -: b2)
   holds LSeg(b1 -: b2,b3) = LSeg(b1,b3);

:: SPPOL_2:th 10
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 natural set
      st b2 in proj2 b1
   holds LSeg(b1 :- b2,b3 + 1) = LSeg(b1,b3 + (b2 .. b1));

:: SPPOL_2:th 11
theorem
L~ <*> the carrier of TOP-REAL 2 = {};

:: SPPOL_2:th 12
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   L~ <*b1*> = {};

:: SPPOL_2:th 13
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1
   holds ex b3 being Element of NAT st
      1 <= b3 & b3 + 1 <= len b1 & b2 in LSeg(b1,b3);

:: SPPOL_2:th 14
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1
   holds ex b3 being Element of NAT st
      1 <= b3 &
       b3 + 1 <= len b1 &
       b2 in LSeg(b1 /. b3,b1 /. (b3 + 1));

:: SPPOL_2:th 15
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 1 <= b3 &
         b3 + 1 <= len b1 &
         b2 in LSeg(b1 /. b3,b1 /. (b3 + 1))
   holds b2 in L~ b1;

:: SPPOL_2:th 16
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
   holds LSeg(b1 /. b2,b1 /. (b2 + 1)) c= L~ b1;

:: SPPOL_2:th 17
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 b2 in LSeg(b1,b3)
   holds b2 in L~ b1;

:: SPPOL_2:th 18
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
      st 2 <= len b1
   holds proj2 b1 c= L~ b1;

:: SPPOL_2:th 19
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b1 is not empty
   holds L~ (b1 ^ <*b2*>) = (L~ b1) \/ LSeg(b1 /. len b1,b2);

:: SPPOL_2:th 20
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b1 is not empty
   holds L~ (<*b2*> ^ b1) = (LSeg(b2,b1 /. 1)) \/ L~ b1;

:: SPPOL_2:th 21
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
L~ <*b1,b2*> = LSeg(b1,b2);

:: SPPOL_2:th 22
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
   L~ b1 = L~ Rev b1;

:: SPPOL_2:th 23
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
      st b1 is not empty & b2 is not empty
   holds L~ (b1 ^ b2) = ((L~ b1) \/ LSeg(b1 /. len b1,b2 /. 1)) \/ L~ b2;

:: SPPOL_2: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
      st b2 in proj2 b1
   holds L~ b1 = (L~ (b1 -: b2)) \/ L~ (b1 :- b2);

:: SPPOL_2: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 b2 in LSeg(b1,b3)
   holds L~ b1 = L~ Ins(b1,b3,b2);

:: SPPOL_2:exreg 1
registration
  cluster Relation-like Function-like finite FinSequence-like being_S-Seq FinSequence of the carrier of TOP-REAL 2;
end;

:: SPPOL_2:condreg 1
registration
  cluster being_S-Seq -> one-to-one non trivial special unfolded s.n.c. (FinSequence of the carrier of TOP-REAL 2);
end;

:: SPPOL_2:condreg 2
registration
  cluster one-to-one non trivial special unfolded s.n.c. -> being_S-Seq (FinSequence of the carrier of TOP-REAL 2);
end;

:: SPPOL_2:condreg 3
registration
  cluster being_S-Seq -> non empty (FinSequence of the carrier of TOP-REAL 2);
end;

:: SPPOL_2:exreg 2
registration
  cluster Relation-like Function-like one-to-one finite FinSequence-like non trivial special unfolded s.n.c. FinSequence of the carrier of TOP-REAL 2;
end;

:: SPPOL_2:th 27
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
      st len b1 <= 2
   holds b1 is unfolded;

:: SPPOL_2:funcreg 1
registration
  let a1 be unfolded FinSequence of the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  cluster a1 | a2 -> unfolded;
end;

:: SPPOL_2:funcreg 2
registration
  let a1 be unfolded FinSequence of the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  cluster a1 /^ a2 -> unfolded;
end;

:: SPPOL_2:th 28
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in proj2 b1 & b1 is unfolded
   holds b1 :- b2 is unfolded;

:: SPPOL_2:funcreg 3
registration
  let a1 be unfolded FinSequence of the carrier of TOP-REAL 2;
  let a2 be Element of the carrier of TOP-REAL 2;
  cluster a1 -: a2 -> unfolded;
end;

:: SPPOL_2:th 29
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
      st b1 is unfolded
   holds Rev b1 is unfolded;

:: SPPOL_2:th 30
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b1 is unfolded &
         (LSeg(b2,b1 /. 1)) /\ LSeg(b1,1) = {b1 /. 1}
   holds <*b2*> ^ b1 is unfolded;

:: SPPOL_2:th 31
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 unfolded &
         b3 + 1 = len b1 &
         (LSeg(b1,b3)) /\ LSeg(b1 /. len b1,b2) = {b1 /. len b1}
   holds b1 ^ <*b2*> is unfolded;

:: SPPOL_2:th 32
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of NAT
      st b1 is unfolded &
         b2 is unfolded &
         b3 + 1 = len b1 &
         (LSeg(b1,b3)) /\ LSeg(b1 /. len b1,b2 /. 1) = {b1 /. len b1} &
         (LSeg(b1 /. len b1,b2 /. 1)) /\ LSeg(b2,1) = {b2 /. 1}
   holds b1 ^ b2 is unfolded;

:: SPPOL_2:th 33
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 unfolded & b2 in LSeg(b1,b3)
   holds Ins(b1,b3,b2) is unfolded;

:: SPPOL_2:th 34
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
      st len b1 <= 2
   holds b1 is s.n.c.;

:: SPPOL_2:funcreg 4
registration
  let a1 be s.n.c. FinSequence of the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  cluster a1 | a2 -> s.n.c.;
end;

:: SPPOL_2:funcreg 5
registration
  let a1 be s.n.c. FinSequence of the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  cluster a1 /^ a2 -> s.n.c.;
end;

:: SPPOL_2:funcreg 6
registration
  let a1 be s.n.c. FinSequence of the carrier of TOP-REAL 2;
  let a2 be Element of the carrier of TOP-REAL 2;
  cluster a1 -: a2 -> s.n.c.;
end;

:: SPPOL_2:th 35
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in proj2 b1 & b1 is s.n.c.
   holds b1 :- b2 is s.n.c.;

:: SPPOL_2:th 36
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
      st b1 is s.n.c.
   holds Rev b1 is s.n.c.;

:: SPPOL_2:th 37
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
      st b1 is s.n.c. &
         b2 is s.n.c. &
         L~ b1 misses L~ b2 &
         (for b3 being Element of NAT
               st 1 <= b3 & b3 + 2 <= len b1
            holds LSeg(b1,b3) misses LSeg(b1 /. len b1,b2 /. 1)) &
         (for b3 being Element of NAT
               st 2 <= b3 & b3 + 1 <= len b2
            holds LSeg(b2,b3) misses LSeg(b1 /. len b1,b2 /. 1))
   holds b1 ^ b2 is s.n.c.;

:: SPPOL_2:th 38
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 unfolded & b1 is s.n.c. & b2 in LSeg(b1,b3) & not b2 in proj2 b1
   holds Ins(b1,b3,b2) is s.n.c.;

:: SPPOL_2:funcreg 7
registration
  cluster <*> the carrier of TOP-REAL 2 -> special;
end;

:: SPPOL_2:th 39
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
   <*b1*> is special;

:: SPPOL_2:th 40
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st (b1 `1 = b2 `1 or b1 `2 = b2 `2)
   holds <*b1,b2*> is special;

:: SPPOL_2:funcreg 8
registration
  let a1 be special FinSequence of the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  cluster a1 | a2 -> special;
end;

:: SPPOL_2:funcreg 9
registration
  let a1 be special FinSequence of the carrier of TOP-REAL 2;
  let a2 be Element of NAT;
  cluster a1 /^ a2 -> special;
end;

:: SPPOL_2:th 41
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in proj2 b1 & b1 is special
   holds b1 :- b2 is special;

:: SPPOL_2:funcreg 10
registration
  let a1 be special FinSequence of the carrier of TOP-REAL 2;
  let a2 be Element of the carrier of TOP-REAL 2;
  cluster a1 -: a2 -> special;
end;

:: SPPOL_2:th 42
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
      st b1 is special
   holds Rev b1 is special;

:: SPPOL_2:th 44
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 special & b2 in LSeg(b1,b3)
   holds Ins(b1,b3,b2) is special;

:: SPPOL_2:th 45
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in proj2 b1 & 1 <> b2 .. b1 & b2 .. b1 <> len b1 & b1 is unfolded & b1 is s.n.c.
   holds (L~ (b1 -: b2)) /\ L~ (b1 :- b2) = {b2};

:: SPPOL_2:th 46
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 <> b2 &
         (b1 `1 = b2 `1 or b1 `2 = b2 `2)
   holds <*b1,b2*> is being_S-Seq;

:: SPPOL_2:modenot 1
definition
  mode S-Sequence_in_R2 is being_S-Seq FinSequence of the carrier of TOP-REAL 2;
end;

:: SPPOL_2:th 47
theorem
for b1 being being_S-Seq FinSequence of the carrier of TOP-REAL 2 holds
   Rev b1 is being_S-Seq;

:: SPPOL_2:th 48
theorem
for b1 being Element of NAT
for b2 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
      st b1 in dom b2
   holds b2 /. b1 in L~ b2;

:: SPPOL_2:th 49
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 <> b2 &
         (b1 `1 = b2 `1 or b1 `2 = b2 `2)
   holds LSeg(b1,b2) is being_S-P_arc;

:: SPPOL_2:th 50
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
      st b1 in proj2 b2 & b1 .. b2 <> 1
   holds b2 -: b1 is being_S-Seq;

:: SPPOL_2:th 51
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
      st b1 in proj2 b2 & b1 .. b2 <> len b2
   holds b2 :- b1 is being_S-Seq;

:: SPPOL_2:th 52
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of NAT
for b3 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
      st b1 in LSeg(b3,b2) & not b1 in proj2 b3
   holds Ins(b3,b2,b1) is being_S-Seq;

:: SPPOL_2:exreg 3
registration
  cluster being_S-P_arc Element of bool the carrier of TOP-REAL 2;
end;

:: SPPOL_2:th 53
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_S-P_arc_joining b2,b3
   holds b1 is_S-P_arc_joining b3,b2;

:: SPPOL_2:prednot 1 => SPPOL_2:pred 1
definition
  let a1, a2 be Element of the carrier of TOP-REAL 2;
  let a3 be Element of bool the carrier of TOP-REAL 2;
  pred A1,A2 split A3 means
    a1 <> a2 &
     (ex b1, b2 being being_S-Seq FinSequence of the carrier of TOP-REAL 2 st
        a1 = b1 /. 1 &
         a1 = b2 /. 1 &
         a2 = b1 /. len b1 &
         a2 = b2 /. len b2 &
         (L~ b1) /\ L~ b2 = {a1,a2} &
         a3 = (L~ b1) \/ L~ b2);
end;

:: SPPOL_2:dfs 1
definiens
  let a1, a2 be Element of the carrier of TOP-REAL 2;
  let a3 be Element of bool the carrier of TOP-REAL 2;
To prove
     a1,a2 split a3
it is sufficient to prove
  thus a1 <> a2 &
     (ex b1, b2 being being_S-Seq FinSequence of the carrier of TOP-REAL 2 st
        a1 = b1 /. 1 &
         a1 = b2 /. 1 &
         a2 = b1 /. len b1 &
         a2 = b2 /. len b2 &
         (L~ b1) /\ L~ b2 = {a1,a2} &
         a3 = (L~ b1) \/ L~ b2);

:: SPPOL_2:def 1
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of bool the carrier of TOP-REAL 2 holds
      b1,b2 split b3
   iff
      b1 <> b2 &
       (ex b4, b5 being being_S-Seq FinSequence of the carrier of TOP-REAL 2 st
          b1 = b4 /. 1 &
           b1 = b5 /. 1 &
           b2 = b4 /. len b4 &
           b2 = b5 /. len b5 &
           (L~ b4) /\ L~ b5 = {b1,b2} &
           b3 = (L~ b4) \/ L~ b5);

:: SPPOL_2:th 54
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 b2,b3 split b1
   holds b3,b2 split b1;

:: SPPOL_2:th 55
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 b2,b3 split b1 & b4 in b1 & b4 <> b2
   holds b2,b4 split b1;

:: SPPOL_2:th 56
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 b2,b3 split b1 & b4 in b1 & b4 <> b3
   holds b4,b3 split b1;

:: SPPOL_2:th 57
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 b2,b3 split b1 & b4 in b1 & b5 in b1 & b4 <> b5
   holds b4,b5 split b1;

:: SPPOL_2:attrnot 1 => TOPREAL4:attr 1
notation
  let a1 be Element of bool the carrier of TOP-REAL 2;
  synonym special_polygonal for being_special_polygon;
end;

:: SPPOL_2:attrnot 2 => TOPREAL4:attr 1
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  attr a1 is special_polygonal means
    ex b1, b2 being Element of the carrier of TOP-REAL 2 st
       b1,b2 split a1;
end;

:: SPPOL_2:dfs 2
definiens
  let a1 be Element of bool the carrier of TOP-REAL 2;
To prove
     a1 is being_special_polygon
it is sufficient to prove
  thus ex b1, b2 being Element of the carrier of TOP-REAL 2 st
       b1,b2 split a1;

:: SPPOL_2:def 2
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
      b1 is being_special_polygon
   iff
      ex b2, b3 being Element of the carrier of TOP-REAL 2 st
         b2,b3 split b1;

:: SPPOL_2:funcnot 1 => SPPOL_2:func 1
definition
  let a1, a2, a3, a4 be real set;
  func [.A1,A2,A3,A4.] -> Element of bool the carrier of TOP-REAL 2 equals
    ((LSeg(|[a1,a3]|,|[a1,a4]|)) \/ LSeg(|[a1,a4]|,|[a2,a4]|)) \/ ((LSeg(|[a2,a4]|,|[a2,a3]|)) \/ LSeg(|[a2,a3]|,|[a1,a3]|));
end;

:: SPPOL_2:def 3
theorem
for b1, b2, b3, b4 being real set holds
[.b1,b2,b3,b4.] = ((LSeg(|[b1,b3]|,|[b1,b4]|)) \/ LSeg(|[b1,b4]|,|[b2,b4]|)) \/ ((LSeg(|[b2,b4]|,|[b2,b3]|)) \/ LSeg(|[b2,b3]|,|[b1,b3]|));

:: SPPOL_2:funcreg 11
registration
  let a1 be Element of NAT;
  let a2, a3 be Element of the carrier of TOP-REAL a1;
  cluster LSeg(a2,a3) -> compact;
end;

:: SPPOL_2:funcreg 12
registration
  let a1, a2, a3, a4 be real set;
  cluster [.a1,a2,a3,a4.] -> non empty compact;
end;

:: SPPOL_2:th 58
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds [.b1,b2,b3,b4.] = {b5 where b5 is Element of the carrier of TOP-REAL 2: ((b5 `1 = b1 & b5 `2 <= b4 implies b5 `2 < b3) &
    (b5 `1 <= b2 & b1 <= b5 `1 implies b5 `2 <> b4) &
    (b5 `1 <= b2 & b1 <= b5 `1 implies b5 `2 <> b3) implies b5 `1 = b2 & b5 `2 <= b4 & b3 <= b5 `2)};

:: SPPOL_2:th 59
theorem
for b1, b2, b3, b4 being real set
      st b1 <> b2 & b3 <> b4
   holds [.b1,b2,b3,b4.] is being_special_polygon;

:: SPPOL_2:th 60
theorem
R^2-unit_square = [.0,1,0,1.];

:: SPPOL_2:exreg 4
registration
  cluster being_special_polygon Element of bool the carrier of TOP-REAL 2;
end;

:: SPPOL_2:th 61
theorem
R^2-unit_square is being_special_polygon;

:: SPPOL_2:exreg 5
registration
  cluster being_special_polygon Element of bool the carrier of TOP-REAL 2;
end;

:: SPPOL_2:condreg 4
registration
  cluster being_special_polygon -> non empty (Element of bool the carrier of TOP-REAL 2);
end;

:: SPPOL_2:condreg 5
registration
  cluster being_special_polygon -> non trivial (Element of bool the carrier of TOP-REAL 2);
end;

:: SPPOL_2:modenot 2
definition
  mode Special_polygon_in_R2 is being_special_polygon Element of bool the carrier of TOP-REAL 2;
end;

:: SPPOL_2:th 62
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st b1 is being_S-P_arc
   holds b1 is compact(TOP-REAL 2);

:: SPPOL_2:th 63
theorem
for b1 being being_special_polygon Element of bool the carrier of TOP-REAL 2 holds
   b1 is compact(TOP-REAL 2);

:: SPPOL_2:th 64
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
   st b1 is being_special_polygon
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b2 in b1 & b3 in b1
   holds b2,b3 split b1;

:: SPPOL_2:th 65
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
   st b1 is being_special_polygon
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b2 in b1 & b3 in b1
   holds ex b4, b5 being Element of bool the carrier of TOP-REAL 2 st
      b4 is_S-P_arc_joining b2,b3 & b5 is_S-P_arc_joining b2,b3 & b4 /\ b5 = {b2,b3} & b1 = b4 \/ b5;