Article TOPREAL4, MML version 4.99.1005

:: TOPREAL4:prednot 1 => TOPREAL4:pred 1
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  let a2, a3 be Element of the carrier of TOP-REAL 2;
  pred A1 is_S-P_arc_joining A2,A3 means
    ex b1 being FinSequence of the carrier of TOP-REAL 2 st
       b1 is being_S-Seq & a1 = L~ b1 & a2 = b1 /. 1 & a3 = b1 /. len b1;
end;

:: TOPREAL4:dfs 1
definiens
  let a1 be Element of bool the carrier of TOP-REAL 2;
  let a2, a3 be Element of the carrier of TOP-REAL 2;
To prove
     a1 is_S-P_arc_joining a2,a3
it is sufficient to prove
  thus ex b1 being FinSequence of the carrier of TOP-REAL 2 st
       b1 is being_S-Seq & a1 = L~ b1 & a2 = b1 /. 1 & a3 = b1 /. len b1;

:: TOPREAL4:def 1
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 holds
   b1 is_S-P_arc_joining b2,b3
iff
   ex b4 being FinSequence of the carrier of TOP-REAL 2 st
      b4 is being_S-Seq & b1 = L~ b4 & b2 = b4 /. 1 & b3 = b4 /. len b4;

:: TOPREAL4:attrnot 1 => TOPREAL4:attr 1
definition
  let a1 be Element of bool the carrier of TOP-REAL 2;
  attr a1 is being_special_polygon means
    ex b1, b2 being Element of the carrier of TOP-REAL 2 st
       ex b3, b4 being Element of bool the carrier of TOP-REAL 2 st
          b1 <> b2 & b1 in a1 & b2 in a1 & b3 is_S-P_arc_joining b1,b2 & b4 is_S-P_arc_joining b1,b2 & b3 /\ b4 = {b1,b2} & a1 = b3 \/ b4;
end;

:: TOPREAL4: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
       ex b3, b4 being Element of bool the carrier of TOP-REAL 2 st
          b1 <> b2 & b1 in a1 & b2 in a1 & b3 is_S-P_arc_joining b1,b2 & b4 is_S-P_arc_joining b1,b2 & b3 /\ b4 = {b1,b2} & a1 = b3 \/ b4;

:: TOPREAL4: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
         ex b4, b5 being Element of bool the carrier of TOP-REAL 2 st
            b2 <> b3 & b2 in b1 & b3 in b1 & b4 is_S-P_arc_joining b2,b3 & b5 is_S-P_arc_joining b2,b3 & b4 /\ b5 = {b2,b3} & b1 = b4 \/ b5;

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

:: TOPREAL4:attrnot 2 => TOPREAL4:attr 2
definition
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  attr a2 is being_Region means
    a2 is open(a1) & a2 is connected(a1);
end;

:: TOPREAL4:dfs 3
definiens
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is being_Region
it is sufficient to prove
  thus a2 is open(a1) & a2 is connected(a1);

:: TOPREAL4:def 3
theorem
for b1 being TopStruct
for b2 being Element of bool the carrier of b1 holds
      b2 is being_Region(b1)
   iff
      b2 is open(b1) & b2 is connected(b1);

:: TOPREAL4:prednot 3 => TOPREAL4:attr 2
notation
  let a1 be TopStruct;
  let a2 be Element of bool the carrier of a1;
  synonym a2 is_Region for being_Region;
end;

:: TOPREAL4:th 2
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 being_S-P_arc;

:: TOPREAL4:th 3
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_an_arc_of b2,b3;

:: TOPREAL4:th 4
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 b2 in b1 & b3 in b1;

:: TOPREAL4:th 5
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 b2 <> b3;

:: TOPREAL4:th 6
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st b1 is being_special_polygon
   holds b1 is being_simple_closed_curve;

:: TOPREAL4:th 7
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being FinSequence of the carrier of TOP-REAL 2
for b4 being Element of REAL
for b5 being Element of the carrier of Euclid 2
      st b1 `1 = b2 `1 &
         b1 `2 <> b2 `2 &
         b1 in Ball(b5,b4) &
         b2 in Ball(b5,b4) &
         b3 = <*b1,|[b1 `1,(b1 `2 + (b2 `2)) / 2]|,b2*>
   holds b3 is being_S-Seq & b3 /. 1 = b1 & b3 /. len b3 = b2 & L~ b3 is_S-P_arc_joining b1,b2 & L~ b3 c= Ball(b5,b4);

:: TOPREAL4:th 8
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being FinSequence of the carrier of TOP-REAL 2
for b4 being Element of REAL
for b5 being Element of the carrier of Euclid 2
      st b1 `1 <> b2 `1 &
         b1 `2 = b2 `2 &
         b1 in Ball(b5,b4) &
         b2 in Ball(b5,b4) &
         b3 = <*b1,|[(b1 `1 + (b2 `1)) / 2,b1 `2]|,b2*>
   holds b3 is being_S-Seq & b3 /. 1 = b1 & b3 /. len b3 = b2 & L~ b3 is_S-P_arc_joining b1,b2 & L~ b3 c= Ball(b5,b4);

:: TOPREAL4:th 9
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being FinSequence of the carrier of TOP-REAL 2
for b4 being Element of REAL
for b5 being Element of the carrier of Euclid 2
      st b1 `1 <> b2 `1 &
         b1 `2 <> b2 `2 &
         b1 in Ball(b5,b4) &
         b2 in Ball(b5,b4) &
         |[b1 `1,b2 `2]| in Ball(b5,b4) &
         b3 = <*b1,|[b1 `1,b2 `2]|,b2*>
   holds b3 is being_S-Seq & b3 /. 1 = b1 & b3 /. len b3 = b2 & L~ b3 is_S-P_arc_joining b1,b2 & L~ b3 c= Ball(b5,b4);

:: TOPREAL4:th 10
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being FinSequence of the carrier of TOP-REAL 2
for b4 being Element of REAL
for b5 being Element of the carrier of Euclid 2
      st b1 `1 <> b2 `1 &
         b1 `2 <> b2 `2 &
         b1 in Ball(b5,b4) &
         b2 in Ball(b5,b4) &
         |[b2 `1,b1 `2]| in Ball(b5,b4) &
         b3 = <*b1,|[b2 `1,b1 `2]|,b2*>
   holds b3 is being_S-Seq & b3 /. 1 = b1 & b3 /. len b3 = b2 & L~ b3 is_S-P_arc_joining b1,b2 & L~ b3 c= Ball(b5,b4);

:: TOPREAL4:th 11
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of REAL
for b4 being Element of the carrier of Euclid 2
      st b1 <> b2 & b1 in Ball(b4,b3) & b2 in Ball(b4,b3)
   holds ex b5 being Element of bool the carrier of TOP-REAL 2 st
      b5 is_S-P_arc_joining b1,b2 & b5 c= Ball(b4,b3);

:: TOPREAL4:th 12
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2, b3 being FinSequence of the carrier of TOP-REAL 2
      st b1 <> b2 /. 1 &
         (b2 /. 1) `2 = b1 `2 &
         b2 is being_S-Seq &
         b1 in LSeg(b2,1) &
         b3 = <*b2 /. 1,|[((b2 /. 1) `1 + (b1 `1)) / 2,(b2 /. 1) `2]|,b1*>
   holds b3 is being_S-Seq &
    b3 /. 1 = b2 /. 1 &
    b3 /. len b3 = b1 &
    L~ b3 is_S-P_arc_joining b2 /. 1,b1 &
    L~ b3 c= L~ b2 &
    L~ b3 = (L~ (b2 | 1)) \/ LSeg(b2 /. 1,b1);

:: TOPREAL4:th 13
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2, b3 being FinSequence of the carrier of TOP-REAL 2
      st b1 <> b2 /. 1 &
         (b2 /. 1) `1 = b1 `1 &
         b2 is being_S-Seq &
         b1 in LSeg(b2,1) &
         b3 = <*b2 /. 1,|[(b2 /. 1) `1,((b2 /. 1) `2 + (b1 `2)) / 2]|,b1*>
   holds b3 is being_S-Seq &
    b3 /. 1 = b2 /. 1 &
    b3 /. len b3 = b1 &
    L~ b3 is_S-P_arc_joining b2 /. 1,b1 &
    L~ b3 c= L~ b2 &
    L~ b3 = (L~ (b2 | 1)) \/ LSeg(b2 /. 1,b1);

:: TOPREAL4:th 14
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2, b3 being FinSequence of the carrier of TOP-REAL 2
for b4 being Element of NAT
      st b1 <> b2 /. 1 & b2 is being_S-Seq & b4 in dom b2 & b4 + 1 in dom b2 & 1 < b4 & b1 in LSeg(b2,b4) & b1 <> b2 /. b4 & b1 <> b2 /. (b4 + 1) & b3 = (b2 | b4) ^ <*b1*>
   holds b3 is being_S-Seq &
    b3 /. 1 = b2 /. 1 &
    b3 /. len b3 = b1 &
    L~ b3 is_S-P_arc_joining b2 /. 1,b1 &
    L~ b3 c= L~ b2 &
    L~ b3 = (L~ (b2 | b4)) \/ LSeg(b2 /. b4,b1);

:: TOPREAL4:th 15
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
      st b1 /. 2 <> b1 /. 1 &
         b1 is being_S-Seq &
         (b1 /. 2) `2 = (b1 /. 1) `2 &
         b2 = <*b1 /. 1,|[((b1 /. 1) `1 + ((b1 /. 2) `1)) / 2,(b1 /. 1) `2]|,b1 /. 2*>
   holds b2 is being_S-Seq &
    b2 /. 1 = b1 /. 1 &
    b2 /. len b2 = b1 /. 2 &
    L~ b2 is_S-P_arc_joining b1 /. 1,b1 /. 2 &
    L~ b2 c= L~ b1 &
    L~ b2 = (L~ (b1 | 1)) \/ LSeg(b1 /. 1,b1 /. 2) &
    L~ b2 = (L~ (b1 | 2)) \/ LSeg(b1 /. 2,b1 /. 2);

:: TOPREAL4:th 16
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
      st b1 /. 2 <> b1 /. 1 &
         b1 is being_S-Seq &
         (b1 /. 2) `1 = (b1 /. 1) `1 &
         b2 = <*b1 /. 1,|[(b1 /. 1) `1,((b1 /. 1) `2 + ((b1 /. 2) `2)) / 2]|,b1 /. 2*>
   holds b2 is being_S-Seq &
    b2 /. 1 = b1 /. 1 &
    b2 /. len b2 = b1 /. 2 &
    L~ b2 is_S-P_arc_joining b1 /. 1,b1 /. 2 &
    L~ b2 c= L~ b1 &
    L~ b2 = (L~ (b1 | 1)) \/ LSeg(b1 /. 1,b1 /. 2) &
    L~ b2 = (L~ (b1 | 2)) \/ LSeg(b1 /. 2,b1 /. 2);

:: TOPREAL4:th 17
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of NAT
      st b1 /. b3 <> b1 /. 1 & b1 is being_S-Seq & 2 < b3 & b3 in dom b1 & b2 = b1 | b3
   holds b2 is being_S-Seq &
    b2 /. 1 = b1 /. 1 &
    b2 /. len b2 = b1 /. b3 &
    L~ b2 is_S-P_arc_joining b1 /. 1,b1 /. b3 &
    L~ b2 c= L~ b1 &
    L~ b2 = (L~ (b1 | b3)) \/ LSeg(b1 /. b3,b1 /. b3);

:: TOPREAL4:th 18
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of NAT
      st b1 <> b2 /. 1 & b2 is being_S-Seq & b1 in LSeg(b2,b3)
   holds ex b4 being FinSequence of the carrier of TOP-REAL 2 st
      b4 is being_S-Seq &
       b4 /. 1 = b2 /. 1 &
       b4 /. len b4 = b1 &
       L~ b4 is_S-P_arc_joining b2 /. 1,b1 &
       L~ b4 c= L~ b2 &
       L~ b4 = (L~ (b2 | b3)) \/ LSeg(b2 /. b3,b1);

:: TOPREAL4:th 19
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
      st b1 <> b2 /. 1 & b2 is being_S-Seq & b1 in L~ b2
   holds ex b3 being FinSequence of the carrier of TOP-REAL 2 st
      b3 is being_S-Seq & b3 /. 1 = b2 /. 1 & b3 /. len b3 = b1 & L~ b3 is_S-P_arc_joining b2 /. 1,b1 & L~ b3 c= L~ b2;

:: TOPREAL4:th 20
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2, b3 being FinSequence of the carrier of TOP-REAL 2
for b4 being Element of REAL
for b5 being Element of the carrier of Euclid 2
      st (b1 `1 = (b2 /. len b2) `1 & b1 `2 <> (b2 /. len b2) `2 or b1 `1 <> (b2 /. len b2) `1 & b1 `2 = (b2 /. len b2) `2) &
         not b2 /. 1 in Ball(b5,b4) &
         b2 /. len b2 in Ball(b5,b4) &
         b1 in Ball(b5,b4) &
         b2 is being_S-Seq &
         (LSeg(b2 /. len b2,b1)) /\ L~ b2 = {b2 /. len b2} &
         b3 = b2 ^ <*b1*>
   holds b3 is being_S-Seq & L~ b3 is_S-P_arc_joining b2 /. 1,b1 & L~ b3 c= (L~ b2) \/ Ball(b5,b4);

:: TOPREAL4:th 21
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2, b3 being FinSequence of the carrier of TOP-REAL 2
for b4 being Element of REAL
for b5 being Element of the carrier of Euclid 2
      st not b2 /. 1 in Ball(b5,b4) &
         b2 /. len b2 in Ball(b5,b4) &
         b1 in Ball(b5,b4) &
         |[b1 `1,(b2 /. len b2) `2]| in Ball(b5,b4) &
         b2 is being_S-Seq &
         b1 `1 <> (b2 /. len b2) `1 &
         b1 `2 <> (b2 /. len b2) `2 &
         b3 = b2 ^ <*|[b1 `1,(b2 /. len b2) `2]|,b1*> &
         ((LSeg(b2 /. len b2,|[b1 `1,(b2 /. len b2) `2]|)) \/ LSeg(|[b1 `1,(b2 /. len b2) `2]|,b1)) /\ L~ b2 = {b2 /. len b2}
   holds L~ b3 is_S-P_arc_joining b2 /. 1,b1 & L~ b3 c= (L~ b2) \/ Ball(b5,b4);

:: TOPREAL4:th 22
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2, b3 being FinSequence of the carrier of TOP-REAL 2
for b4 being Element of REAL
for b5 being Element of the carrier of Euclid 2
      st not b2 /. 1 in Ball(b5,b4) &
         b2 /. len b2 in Ball(b5,b4) &
         b1 in Ball(b5,b4) &
         |[(b2 /. len b2) `1,b1 `2]| in Ball(b5,b4) &
         b2 is being_S-Seq &
         b1 `1 <> (b2 /. len b2) `1 &
         b1 `2 <> (b2 /. len b2) `2 &
         b3 = b2 ^ <*|[(b2 /. len b2) `1,b1 `2]|,b1*> &
         ((LSeg(b2 /. len b2,|[(b2 /. len b2) `1,b1 `2]|)) \/ LSeg(|[(b2 /. len b2) `1,b1 `2]|,b1)) /\ L~ b2 = {b2 /. len b2}
   holds L~ b3 is_S-P_arc_joining b2 /. 1,b1 & L~ b3 c= (L~ b2) \/ Ball(b5,b4);

:: TOPREAL4:th 23
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of REAL
for b4 being Element of the carrier of Euclid 2
      st not b2 /. 1 in Ball(b4,b3) & b2 /. len b2 in Ball(b4,b3) & b1 in Ball(b4,b3) & b2 is being_S-Seq & not b1 in L~ b2
   holds ex b5 being FinSequence of the carrier of TOP-REAL 2 st
      L~ b5 is_S-P_arc_joining b2 /. 1,b1 & L~ b5 c= (L~ b2) \/ Ball(b4,b3);

:: TOPREAL4:th 24
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
for b5 being Element of bool the carrier of TOP-REAL 2
for b6 being Element of REAL
for b7 being Element of the carrier of Euclid 2
      st b2 <> b3 & b5 is_S-P_arc_joining b3,b4 & b5 c= b1 & b2 in Ball(b7,b6) & b4 in Ball(b7,b6) & Ball(b7,b6) c= b1
   holds ex b8 being Element of bool the carrier of TOP-REAL 2 st
      b8 is_S-P_arc_joining b3,b2 & b8 c= b1;

:: TOPREAL4:th 25
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
      st b1 is being_Region(TOP-REAL 2) &
         b2 = {b4 where b4 is Element of the carrier of TOP-REAL 2: b4 <> b3 &
          b4 in b1 &
          (for b5 being Element of bool the carrier of TOP-REAL 2
                st b5 is_S-P_arc_joining b3,b4
             holds not b5 c= b1)}
   holds b2 is open(TOP-REAL 2);

:: TOPREAL4:th 26
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2, b3 being Element of bool the carrier of TOP-REAL 2
      st b2 is being_Region(TOP-REAL 2) &
         b1 in b2 &
         b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: (b4 <> b1 implies ex b5 being Element of bool the carrier of TOP-REAL 2 st
            b5 is_S-P_arc_joining b1,b4 & b5 c= b2)}
   holds b3 is open(TOP-REAL 2);

:: TOPREAL4:th 27
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2, b3 being Element of bool the carrier of TOP-REAL 2
      st b1 in b2 &
         b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: (b4 <> b1 implies ex b5 being Element of bool the carrier of TOP-REAL 2 st
            b5 is_S-P_arc_joining b1,b4 & b5 c= b2)}
   holds b3 c= b2;

:: TOPREAL4:th 28
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2, b3 being Element of bool the carrier of TOP-REAL 2
      st b2 is being_Region(TOP-REAL 2) &
         b1 in b2 &
         b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: (b4 <> b1 implies ex b5 being Element of bool the carrier of TOP-REAL 2 st
            b5 is_S-P_arc_joining b1,b4 & b5 c= b2)}
   holds b2 c= b3;

:: TOPREAL4:th 29
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2, b3 being Element of bool the carrier of TOP-REAL 2
      st b2 is being_Region(TOP-REAL 2) &
         b1 in b2 &
         b3 = {b4 where b4 is Element of the carrier of TOP-REAL 2: (b4 <> b1 implies ex b5 being Element of bool the carrier of TOP-REAL 2 st
            b5 is_S-P_arc_joining b1,b4 & b5 c= b2)}
   holds b2 = b3;

:: TOPREAL4:th 30
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
      st b3 is being_Region(TOP-REAL 2) & b1 in b3 & b2 in b3 & b1 <> b2
   holds ex b4 being Element of bool the carrier of TOP-REAL 2 st
      b4 is_S-P_arc_joining b1,b2 & b4 c= b3;