Article GOBRD12, MML version 4.99.1005

:: GOBRD12:th 2
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 NAT
      st b2 <= len GoB b1 & b3 <= width GoB b1
   holds Int cell(GoB b1,b2,b3) c= (L~ b1) `;

:: GOBRD12:th 3
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 NAT
      st b2 <= len GoB b1 & b3 <= width GoB b1
   holds Cl Down(Int cell(GoB b1,b2,b3),(L~ b1) `) = (cell(GoB b1,b2,b3)) /\ ((L~ b1) `);

:: GOBRD12:th 4
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 NAT
      st b2 <= len GoB b1 & b3 <= width GoB b1
   holds Cl Down(Int cell(GoB b1,b2,b3),(L~ b1) `) is connected((TOP-REAL 2) | ((L~ b1) `)) &
    Down(Int cell(GoB b1,b2,b3),(L~ b1) `) = Int cell(GoB b1,b2,b3);

:: GOBRD12:th 5
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) ` = union {Cl Down(Int cell(GoB b1,b2,b3),(L~ b1) `) where b2 is Element of NAT, b3 is Element of NAT: b2 <= len GoB b1 & b3 <= width GoB b1};

:: GOBRD12:th 6
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   (Down(LeftComp b1,(L~ b1) `)) \/ Down(RightComp b1,(L~ b1) `) is a_union_of_components of (TOP-REAL 2) | ((L~ b1) `) &
    Down(LeftComp b1,(L~ b1) `) = LeftComp b1 &
    Down(RightComp b1,(L~ b1) `) = RightComp b1;

:: GOBRD12: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, b3, b4, b5 being Element of NAT
      st b2 <= len GoB b1 & b3 <= width GoB b1 & b4 <= len GoB b1 & b5 <= width GoB b1 & b2,b3,b4,b5 are_adjacent2
   holds    Int cell(GoB b1,b2,b3) c= (LeftComp b1) \/ RightComp b1
   iff
      Int cell(GoB b1,b4,b5) c= (LeftComp b1) \/ RightComp b1;

:: GOBRD12:th 8
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 FinSequence of NAT
   st len b2 = len b3 &
      (ex b4 being Element of NAT st
         b4 in dom b2 &
          Int cell(GoB b1,b2 /. b4,b3 /. b4) c= (LeftComp b1) \/ RightComp b1) &
      (for b4 being Element of NAT
            st 1 <= b4 & b4 < len b2
         holds b2 /. b4,b3 /. b4,b2 /. (b4 + 1),b3 /. (b4 + 1) are_adjacent2) &
      (for b4, b5, b6 being Element of NAT
            st b4 in dom b2 & b5 = b2 . b4 & b6 = b3 . b4
         holds b5 <= len GoB b1 & b6 <= width GoB b1)
for b4 being Element of NAT
      st b4 in dom b2
   holds Int cell(GoB b1,b2 /. b4,b3 /. b4) c= (LeftComp b1) \/ RightComp b1;

:: GOBRD12:th 9
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   ex b2, b3 being Element of NAT st
      b2 <= len GoB b1 &
       b3 <= width GoB b1 &
       Int cell(GoB b1,b2,b3) c= (LeftComp b1) \/ RightComp b1;

:: GOBRD12:th 10
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 NAT
      st b2 <= len GoB b1 & b3 <= width GoB b1
   holds Int cell(GoB b1,b2,b3) c= (LeftComp b1) \/ RightComp b1;

:: GOBRD12:th 11
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) ` = (LeftComp b1) \/ RightComp b1;