Article TOPREAL3, MML version 4.99.1005

:: TOPREAL3:th 6
theorem
for b1, b2, b3 being set holds
1 in dom <*b1,b2,b3*> & 2 in dom <*b1,b2,b3*> & 3 in dom <*b1,b2,b3*>;

:: TOPREAL3:th 7
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
(b1 + b2) `1 = b1 `1 + (b2 `1) &
 (b1 + b2) `2 = b1 `2 + (b2 `2);

:: TOPREAL3:th 8
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
(b1 - b2) `1 = b1 `1 - (b2 `1) &
 (b1 - b2) `2 = b1 `2 - (b2 `2);

:: TOPREAL3:th 9
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being real set holds
   (b2 * b1) `1 = b2 * (b1 `1) & (b2 * b1) `2 = b2 * (b1 `2);

:: TOPREAL3:th 10
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3, b4, b5, b6 being real set
      st b1 = <*b3,b4*> & b2 = <*b5,b6*>
   holds b1 + b2 = <*b3 + b5,b4 + b6*> &
    b1 - b2 = <*b3 - b5,b4 - b6*>;

:: TOPREAL3:th 11
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 `1 = b2 `1 & b1 `2 = b2 `2
   holds b1 = b2;

:: TOPREAL3:th 12
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of Euclid 2
      st b3 = b1 & b4 = b2
   holds (Pitag_dist 2) .(b3,b4) = sqrt ((b1 `1 - (b2 `1)) ^2 + ((b1 `2 - (b2 `2)) ^2));

:: TOPREAL3:th 13
theorem
for b1 being natural set holds
   the carrier of TOP-REAL b1 = the carrier of Euclid b1;

:: TOPREAL3:th 15
theorem
for b1, b2, b3 being real set
      st b1 <= b2
   holds {b4 where b4 is Element of the carrier of TOP-REAL 2: b4 `1 = b3 & b1 <= b4 `2 & b4 `2 <= b2} = LSeg(|[b3,b1]|,|[b3,b2]|);

:: TOPREAL3:th 16
theorem
for b1, b2, b3 being real set
      st b1 <= b2
   holds {b4 where b4 is Element of the carrier of TOP-REAL 2: b4 `2 = b3 & b1 <= b4 `1 & b4 `1 <= b2} = LSeg(|[b1,b3]|,|[b2,b3]|);

:: TOPREAL3:th 17
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2, b3, b4 being real set
      st b1 in LSeg(|[b2,b3]|,|[b2,b4]|)
   holds b1 `1 = b2;

:: TOPREAL3:th 18
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2, b3, b4 being real set
      st b1 in LSeg(|[b2,b3]|,|[b4,b3]|)
   holds b1 `2 = b3;

:: TOPREAL3:th 19
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 `1 <> b2 `1 & b1 `2 = b2 `2
   holds |[(b1 `1 + (b2 `1)) / 2,b1 `2]| in LSeg(b1,b2);

:: TOPREAL3:th 20
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 `1 = b2 `1 & b1 `2 <> b2 `2
   holds |[b1 `1,(b1 `2 + (b2 `2)) / 2]| in LSeg(b1,b2);

:: TOPREAL3:th 21
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being FinSequence of the carrier of TOP-REAL 2
for b5, b6 being natural set
      st b4 = <*b1,b2,b3*> & b5 <> 0 & b5 + 1 < b6
   holds LSeg(b4,b6) = {};

:: TOPREAL3:th 23
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being FinSequence of the carrier of TOP-REAL 2
      st b4 = <*b1,b2,b3*>
   holds L~ b4 = (LSeg(b1,b2)) \/ LSeg(b2,b3);

:: TOPREAL3:th 24
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st b2 in dom b1 & b3 in dom (b1 | b2) & b3 + 1 in dom (b1 | b2)
   holds LSeg(b1,b3) = LSeg(b1 | b2,b3);

:: TOPREAL3:th 25
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of NAT
      st b3 in dom b1 & b3 + 1 in dom b1
   holds LSeg(b1 ^ b2,b3) = LSeg(b1,b3);

:: TOPREAL3:th 26
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL b1
for b3 being natural set holds
   LSeg(b2,b3) c= L~ b2;

:: TOPREAL3:th 27
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   L~ (b1 | b2) c= L~ b1;

:: TOPREAL3:th 28
theorem
for b1 being real set
for b2 being Element of NAT
for b3, b4 being Element of the carrier of TOP-REAL b2
for b5 being Element of the carrier of Euclid b2
      st b3 in Ball(b5,b1) & b4 in Ball(b5,b1)
   holds LSeg(b3,b4) c= Ball(b5,b1);

:: TOPREAL3:th 29
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4, b5, b6, b7, b8 being real set
for b9 being Element of the carrier of Euclid 2
      st b9 = b1 & b1 = |[b4,b5]| & b2 = |[b6,b7]| & b3 = |[b6,b5]| & b2 in Ball(b9,b8)
   holds b3 in Ball(b9,b8);

:: TOPREAL3:th 30
theorem
for b1, b2, b3, b4 being real set
for b5 being Element of the carrier of Euclid 2
      st |[b1,b2]| in Ball(b5,b3) & |[b1,b4]| in Ball(b5,b3)
   holds |[b1,(b2 + b4) / 2]| in Ball(b5,b3);

:: TOPREAL3:th 31
theorem
for b1, b2, b3, b4 being real set
for b5 being Element of the carrier of Euclid 2
      st |[b1,b2]| in Ball(b5,b3) & |[b4,b2]| in Ball(b5,b3)
   holds |[(b1 + b4) / 2,b2]| in Ball(b5,b3);

:: TOPREAL3:th 32
theorem
for b1, b2, b3, b4, b5 being real set
for b6 being Element of the carrier of Euclid 2
      st b1 <> b2 & b3 <> b4 & |[b1,b4]| in Ball(b6,b5) & |[b2,b3]| in Ball(b6,b5) & not |[b1,b3]| in Ball(b6,b5)
   holds |[b2,b4]| in Ball(b6,b5);

:: TOPREAL3:th 33
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being real set
for b3 being Element of the carrier of Euclid 2
for b4 being Element of NAT
      st not b1 /. 1 in Ball(b3,b2) &
         1 <= b4 &
         b4 <= (len b1) - 1 &
         LSeg(b1,b4) meets Ball(b3,b2) &
         (for b5 being Element of NAT
               st 1 <= b5 &
                  b5 <= (len b1) - 1 &
                  (LSeg(b1,b5)) /\ Ball(b3,b2) <> {}
            holds b4 <= b5)
   holds not b1 /. b4 in Ball(b3,b2);

:: TOPREAL3:th 34
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 `2 = b2 `2 & b3 `2 <> b2 `2
   holds ((LSeg(b2,|[b2 `1,b3 `2]|)) \/ LSeg(|[b2 `1,b3 `2]|,b3)) /\ LSeg(b1,b2) = {b2};

:: TOPREAL3:th 35
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 `1 = b2 `1 & b3 `1 <> b2 `1
   holds ((LSeg(b2,|[b3 `1,b2 `2]|)) \/ LSeg(|[b3 `1,b2 `2]|,b3)) /\ LSeg(b1,b2) = {b2};

:: TOPREAL3:th 36
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
(LSeg(b1,|[b1 `1,b2 `2]|)) /\ LSeg(|[b1 `1,b2 `2]|,b2) = {|[b1 `1,b2 `2]|};

:: TOPREAL3:th 37
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2 holds
(LSeg(b1,|[b2 `1,b1 `2]|)) /\ LSeg(|[b2 `1,b1 `2]|,b2) = {|[b2 `1,b1 `2]|};

:: TOPREAL3:th 38
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 `1 = b2 `1 & b1 `2 <> b2 `2
   holds (LSeg(b1,|[b1 `1,(b1 `2 + (b2 `2)) / 2]|)) /\ LSeg(|[b1 `1,(b1 `2 + (b2 `2)) / 2]|,b2) = {|[b1 `1,(b1 `2 + (b2 `2)) / 2]|};

:: TOPREAL3:th 39
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 `1 <> b2 `1 & b1 `2 = b2 `2
   holds (LSeg(b1,|[(b1 `1 + (b2 `1)) / 2,b1 `2]|)) /\ LSeg(|[(b1 `1 + (b2 `1)) / 2,b1 `2]|,b2) = {|[(b1 `1 + (b2 `1)) / 2,b1 `2]|};

:: TOPREAL3:th 40
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 2 < b2 & b2 in dom b1 & b1 is being_S-Seq
   holds b1 | b2 is being_S-Seq;

:: TOPREAL3:th 41
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being FinSequence of the carrier of TOP-REAL 2
      st b1 `1 <> b2 `1 &
         b1 `2 <> b2 `2 &
         b3 = <*b1,|[b1 `1,b2 `2]|,b2*>
   holds b3 /. 1 = b1 & b3 /. len b3 = b2 & b3 is being_S-Seq;

:: TOPREAL3:th 42
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being FinSequence of the carrier of TOP-REAL 2
      st b1 `1 <> b2 `1 &
         b1 `2 <> b2 `2 &
         b3 = <*b1,|[b2 `1,b1 `2]|,b2*>
   holds b3 /. 1 = b1 & b3 /. len b3 = b2 & b3 is being_S-Seq;

:: TOPREAL3:th 43
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being FinSequence of the carrier of TOP-REAL 2
      st b1 `1 = b2 `1 &
         b1 `2 <> b2 `2 &
         b3 = <*b1,|[b1 `1,(b1 `2 + (b2 `2)) / 2]|,b2*>
   holds b3 /. 1 = b1 & b3 /. len b3 = b2 & b3 is being_S-Seq;

:: TOPREAL3:th 44
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being FinSequence of the carrier of TOP-REAL 2
      st b1 `1 <> b2 `1 &
         b1 `2 = b2 `2 &
         b3 = <*b1,|[(b1 `1 + (b2 `1)) / 2,b1 `2]|,b2*>
   holds b3 /. 1 = b1 & b3 /. len b3 = b2 & b3 is being_S-Seq;

:: TOPREAL3:th 45
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st b2 in dom b1 & b2 + 1 in dom b1
   holds L~ (b1 | (b2 + 1)) = (L~ (b1 | b2)) \/ LSeg(b1 /. b2,b1 /. (b2 + 1));

:: TOPREAL3:th 46
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
   st 2 <= len b2 & not b1 in L~ b2
for b3 being Element of NAT
      st 1 <= b3 & b3 <= len b2
   holds b2 /. b3 <> b1;

:: TOPREAL3:th 47
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being FinSequence of the carrier of TOP-REAL 2
      st b1 <> b2 &
         (LSeg(b1,b2)) /\ L~ b3 = {b1}
   holds not b2 in L~ b3;

:: TOPREAL3:th 48
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 real set
for b4 being Element of the carrier of Euclid 2
for b5 being Element of NAT
      st b2 is being_S-Seq & not b2 /. 1 in Ball(b4,b3) & b1 in Ball(b4,b3) & b2 /. len b2 in LSeg(b2,b5) & 1 <= b5 & b5 + 1 <= len b2 & LSeg(b2,b5) meets Ball(b4,b3)
   holds b5 + 1 = len b2;

:: TOPREAL3:th 49
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being real set
for b5 being Element of the carrier of Euclid 2
      st not b1 in Ball(b5,b4) &
         b2 in Ball(b5,b4) &
         b3 in Ball(b5,b4) &
         not b3 in LSeg(b1,b2) &
         (b2 `1 = b3 `1 & b2 `2 <> b3 `2 or b2 `1 <> b3 `1 & b2 `2 = b3 `2) &
         (b1 `1 = b2 `1 or b1 `2 = b2 `2)
   holds (LSeg(b1,b2)) /\ LSeg(b2,b3) = {b2};

:: TOPREAL3:th 50
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being real set
for b5 being Element of the carrier of Euclid 2
      st not b1 in Ball(b5,b4) &
         b2 in Ball(b5,b4) &
         |[b2 `1,b3 `2]| in Ball(b5,b4) &
         b3 in Ball(b5,b4) &
         not |[b2 `1,b3 `2]| in LSeg(b1,b2) &
         b1 `1 = b2 `1 &
         b2 `1 <> b3 `1 &
         b2 `2 <> b3 `2
   holds ((LSeg(b2,|[b2 `1,b3 `2]|)) \/ LSeg(|[b2 `1,b3 `2]|,b3)) /\ LSeg(b1,b2) = {b2};

:: TOPREAL3:th 51
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being real set
for b5 being Element of the carrier of Euclid 2
      st not b1 in Ball(b5,b4) &
         b2 in Ball(b5,b4) &
         |[b3 `1,b2 `2]| in Ball(b5,b4) &
         b3 in Ball(b5,b4) &
         not |[b3 `1,b2 `2]| in LSeg(b1,b2) &
         b1 `2 = b2 `2 &
         b2 `1 <> b3 `1 &
         b2 `2 <> b3 `2
   holds ((LSeg(b2,|[b3 `1,b2 `2]|)) \/ LSeg(|[b3 `1,b2 `2]|,b3)) /\ LSeg(b1,b2) = {b2};