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;