Article SPRECT_4, MML version 4.99.1005
:: SPRECT_4:th 1
theorem
for b1 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
for b2 being closed Element of bool the carrier of TOP-REAL 2
st L~ b1 meets b2 & not b1 /. 1 in b2
holds (L~ R_Cut(b1,First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2))) /\ b2 = {First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2)};
:: SPRECT_4:th 2
theorem
for b1 being non empty FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b1 is being_S-Seq & b2 = b1 /. len b1
holds L~ L_Cut(b1,b2) = {};
:: SPRECT_4:th 4
theorem
for b1 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of NAT
st 1 <= b3 & b3 < len b1 & b2 in L~ mid(b1,b3,len b1)
holds LE b1 /. b3,b2,L~ b1,b1 /. 1,b1 /. len b1;
:: SPRECT_4:th 5
theorem
for b1 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being Element of NAT
st 1 <= b4 & b4 < len b1 & b2 in LSeg(b1,b4) & b3 in LSeg(b2,b1 /. (b4 + 1))
holds LE b2,b3,L~ b1,b1 /. 1,b1 /. len b1;
:: SPRECT_4:th 6
theorem
for b1 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
for b2 being closed Element of bool the carrier of TOP-REAL 2
st L~ b1 meets b2 & not b1 /. len b1 in b2
holds (L~ L_Cut(b1,Last_Point(L~ b1,b1 /. 1,b1 /. len b1,b2))) /\ b2 = {Last_Point(L~ b1,b1 /. 1,b1 /. len b1,b2)};
:: SPRECT_4: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 holds
LeftComp b1 <> RightComp b1;