:: SPRECT_4 semantic presentation

theorem Th1: :: SPRECT_4:1
for b1 being S-Sequence_in_R2
for b2 being closed Subset of (TOP-REAL 2) holds
( L~ b1 meets b2 & not b1 /. 1 in b2 implies (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)} )
proof end;

theorem Th2: :: SPRECT_4:2
for b1 being non empty FinSequence of (TOP-REAL 2)
for b2 being Point of (TOP-REAL 2) holds
( b1 is_S-Seq & b2 = b1 /. (len b1) implies L~ (L_Cut b1,b2) = {} )
proof end;

theorem Th3: :: SPRECT_4:3
canceled;

theorem Th4: :: SPRECT_4:4
for b1 being S-Sequence_in_R2
for b2 being Point of (TOP-REAL 2)
for b3 being Nat holds
( 1 <= b3 & b3 < len b1 & b2 in L~ (mid b1,b3,(len b1)) implies LE b1 /. b3,b2, L~ b1,b1 /. 1,b1 /. (len b1) )
proof end;

theorem Th5: :: SPRECT_4:5
for b1 being S-Sequence_in_R2
for b2, b3 being Point of (TOP-REAL 2)
for b4 being Nat holds
( 1 <= b4 & b4 < len b1 & b2 in LSeg b1,b4 & b3 in LSeg b2,(b1 /. (b4 + 1)) implies LE b2,b3, L~ b1,b1 /. 1,b1 /. (len b1) )
proof end;

theorem Th6: :: SPRECT_4:6
for b1 being S-Sequence_in_R2
for b2 being closed Subset of (TOP-REAL 2) holds
( L~ b1 meets b2 & not b1 /. (len b1) in b2 implies (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)} )
proof end;

Lemma5: for b1 being non constant standard clockwise_oriented special_circular_sequence holds
not ( b1 /. 1 = N-min (L~ b1) & not LeftComp b1 <> RightComp b1 )
proof end;

Lemma6: for b1 being non constant standard special_circular_sequence holds
not ( b1 /. 1 = N-min (L~ b1) & not LeftComp b1 <> RightComp b1 )
proof end;

theorem Th7: :: SPRECT_4:7
for b1 being non constant standard special_circular_sequence holds
LeftComp b1 <> RightComp b1
proof end;