Article GOBRD14, MML version 4.99.1005

:: GOBRD14:th 7
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of the carrier of Euclid 2
      st b3 = 0.REAL 2 & b2 is_outside_component_of L~ b1
   holds ex b4 being Element of REAL st
      0 < b4 & (Ball(b3,b4)) ` c= b2;

:: GOBRD14:th 8
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL b1
      st L~ b2 <> {}
   holds 2 <= len b2;

:: GOBRD14:th 13
theorem
for b1, b2 being Element of NAT
for b3 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
      st 1 <= b1 & b1 < len b3 & 1 <= b2 & b2 < width b3
   holds cell(b3,b1,b2) = product ((1,2)-->([.(b3 *(b1,1)) `1,(b3 *(b1 + 1,1)) `1.],[.(b3 *(1,b2)) `2,(b3 *(1,b2 + 1)) `2.]));

:: GOBRD14:th 14
theorem
for b1, b2 being Element of NAT
for b3 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
      st 1 <= b1 & b1 < len b3 & 1 <= b2 & b2 < width b3
   holds cell(b3,b1,b2) is compact(TOP-REAL 2);

:: GOBRD14:th 15
theorem
for b1, b2, b3 being Element of NAT
for b4 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
      st [b1,b2] in Indices b4 & [b1 + b3,b2] in Indices b4
   holds dist(b4 *(b1,b2),b4 *(b1 + b3,b2)) = (b4 *(b1 + b3,b2)) `1 - ((b4 *(b1,b2)) `1);

:: GOBRD14:th 16
theorem
for b1, b2, b3 being Element of NAT
for b4 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
      st [b1,b2] in Indices b4 & [b1,b2 + b3] in Indices b4
   holds dist(b4 *(b1,b2),b4 *(b1,b2 + b3)) = (b4 *(b1,b2 + b3)) `2 - ((b4 *(b1,b2)) `2);

:: GOBRD14:th 17
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   3 <= (len Gauge(b2,b1)) -' 1;

:: GOBRD14:th 18
theorem
for b1, b2 being Element of NAT
for b3 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
   st b1 <= b2
for b4, b5 being Element of NAT
      st 2 <= b4 & b4 <= (len Gauge(b3,b1)) - 1 & 2 <= b5 & b5 <= (len Gauge(b3,b1)) - 1
   holds ex b6, b7 being Element of NAT st
      2 <= b6 &
       b6 <= (len Gauge(b3,b2)) - 1 &
       2 <= b7 &
       b7 <= (len Gauge(b3,b2)) - 1 &
       [b6,b7] in Indices Gauge(b3,b2) &
       (Gauge(b3,b1)) *(b4,b5) = (Gauge(b3,b2)) *(b6,b7) &
       b6 = 2 + ((2 |^ (b2 -' b1)) * (b4 -' 2)) &
       b7 = 2 + ((2 |^ (b2 -' b1)) * (b5 -' 2));

:: GOBRD14:th 19
theorem
for b1, b2, b3 being Element of NAT
for b4 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
      st [b1,b2] in Indices Gauge(b4,b3) &
         [b1,b2 + 1] in Indices Gauge(b4,b3)
   holds dist((Gauge(b4,b3)) *(b1,b2),(Gauge(b4,b3)) *(b1,b2 + 1)) = ((N-bound b4) - S-bound b4) / (2 |^ b3);

:: GOBRD14:th 20
theorem
for b1, b2, b3 being Element of NAT
for b4 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
      st [b1,b2] in Indices Gauge(b4,b3) &
         [b1 + 1,b2] in Indices Gauge(b4,b3)
   holds dist((Gauge(b4,b3)) *(b1,b2),(Gauge(b4,b3)) *(b1 + 1,b2)) = ((E-bound b4) - W-bound b4) / (2 |^ b3);

:: GOBRD14:th 21
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2, b3 being real set
      st 0 < b2 & 0 < b3
   holds ex b4 being Element of NAT st
      1 < b4 &
       dist((Gauge(b1,b4)) *(1,1),(Gauge(b1,b4)) *(1,2)) < b2 &
       dist((Gauge(b1,b4)) *(1,1),(Gauge(b1,b4)) *(2,1)) < b3;

:: GOBRD14:th 22
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of (TOP-REAL 2) | ((L~ b1) `)
      st b2 is_a_component_of (TOP-REAL 2) | ((L~ b1) `) &
         b2 <> RightComp b1
   holds b2 = LeftComp b1;

:: GOBRD14:th 23
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of bool the carrier of TOP-REAL 2
      st (L~ b1) ` = b2 \/ b3 &
         b2 misses b3 &
         (for b4, b5 being Element of bool the carrier of (TOP-REAL 2) | ((L~ b1) `)
               st b4 = b2 & b5 = b3
            holds b4 is_a_component_of (TOP-REAL 2) | ((L~ b1) `) &
             b5 is_a_component_of (TOP-REAL 2) | ((L~ b1) `)) &
         (b2 = RightComp b1 implies b3 <> LeftComp b1)
   holds b2 = LeftComp b1 & b3 = RightComp b1;

:: GOBRD14:th 24
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   LeftComp b1 misses RightComp b1;

:: GOBRD14:th 25
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   ((L~ b1) \/ RightComp b1) \/ LeftComp b1 = the carrier of TOP-REAL 2;

:: GOBRD14:th 26
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2 holds
      b2 in L~ b1
   iff
      not b2 in LeftComp b1 & not b2 in RightComp b1;

:: GOBRD14:th 27
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2 holds
      b2 in LeftComp b1
   iff
      not b2 in L~ b1 & not b2 in RightComp b1;

:: GOBRD14:th 28
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2 holds
      b2 in RightComp b1
   iff
      not b2 in L~ b1 & not b2 in LeftComp b1;

:: GOBRD14:th 29
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   L~ b1 = (Cl RightComp b1) \ RightComp b1;

:: GOBRD14:th 30
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   L~ b1 = (Cl LeftComp b1) \ LeftComp b1;

:: GOBRD14:th 31
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   Cl RightComp b1 = (RightComp b1) \/ L~ b1;

:: GOBRD14:th 32
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   Cl LeftComp b1 = (LeftComp b1) \/ L~ b1;

:: GOBRD14:funcreg 1
registration
  let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
  cluster L~ a1 -> Jordan;
end;

:: GOBRD14:th 33
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
      st b1 in RightComp b2
   holds W-bound L~ b2 < b1 `1;

:: GOBRD14:th 34
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
      st b1 in RightComp b2
   holds b1 `1 < E-bound L~ b2;

:: GOBRD14:th 35
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
      st b1 in RightComp b2
   holds b1 `2 < N-bound L~ b2;

:: GOBRD14:th 36
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
      st b1 in RightComp b2
   holds S-bound L~ b2 < b1 `2;

:: GOBRD14:th 37
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
      st b1 in RightComp b3 & b2 in LeftComp b3
   holds LSeg(b1,b2) meets L~ b3;

:: GOBRD14:th 38
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   Cl RightComp SpStSeq b1 = product ((1,2)-->([.W-bound L~ SpStSeq b1,E-bound L~ SpStSeq b1.],[.S-bound L~ SpStSeq b1,N-bound L~ SpStSeq b1.]));

:: GOBRD14:th 39
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2 holds
   proj1 .: Cl RightComp b1 = proj1 .: L~ b1;

:: GOBRD14:th 40
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2 holds
   proj2 .: Cl RightComp b1 = proj2 .: L~ b1;

:: GOBRD14:th 41
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2 holds
   RightComp b1 c= RightComp SpStSeq L~ b1;

:: GOBRD14:th 42
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2 holds
   Cl RightComp b1 is compact(TOP-REAL 2);

:: GOBRD14:th 43
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2 holds
   LeftComp b1 is not Bounded(2);

:: GOBRD14:th 44
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2 holds
   LeftComp b1 is_outside_component_of L~ b1;

:: GOBRD14:th 45
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2 holds
   RightComp b1 is_inside_component_of L~ b1;

:: GOBRD14:th 46
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2 holds
   UBD L~ b1 = LeftComp b1;

:: GOBRD14:th 47
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2 holds
   BDD L~ b1 = RightComp b1;

:: GOBRD14:th 48
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 is_outside_component_of L~ b1
   holds b2 = LeftComp b1;

:: GOBRD14:th 49
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 is_inside_component_of L~ b1
   holds b2 meets RightComp b1;

:: GOBRD14:th 50
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
      st b2 is_inside_component_of L~ b1
   holds b2 = BDD L~ b1;

:: GOBRD14:th 51
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2 holds
   W-bound L~ b1 = W-bound RightComp b1;

:: GOBRD14:th 52
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2 holds
   E-bound L~ b1 = E-bound RightComp b1;

:: GOBRD14:th 53
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2 holds
   N-bound L~ b1 = N-bound RightComp b1;

:: GOBRD14:th 54
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2 holds
   S-bound L~ b1 = S-bound RightComp b1;