reserve n for Nat;

theorem Th36:
  for f be non constant standard special_circular_sequence for p,q
  be Point of TOP-REAL 2 for g be connected Subset of TOP-REAL 2 st p in
  RightComp f & q in LeftComp f & p in g & q in g holds g meets L~f
proof
  let f be non constant standard special_circular_sequence;
  let p,q be Point of TOP-REAL 2;
  let g be connected Subset of TOP-REAL 2;
  assume that
A1: p in RightComp f and
A2: q in LeftComp f and
A3: p in g and
A4: q in g;
  assume g misses L~f;
  then g c= (L~f)` by TDLAT_1:2;
  then reconsider A = g as Subset of (TOP-REAL 2)|(L~f)` by PRE_TOPC:8;
  RightComp f is_a_component_of (L~f)` by GOBOARD9:def 2;
  then consider R being Subset of (TOP-REAL 2)|(L~f)` such that
A5: R = RightComp f and
A6: R is a_component by CONNSP_1:def 6;
  R /\ A <> {} by A1,A3,A5,XBOOLE_0:def 4;
  then
A7: R meets A;
  LeftComp f is_a_component_of (L~f)` by GOBOARD9:def 1;
  then consider L being Subset of (TOP-REAL 2)|(L~f)` such that
A8: L = LeftComp f and
A9: L is a_component by CONNSP_1:def 6;
  L /\ A <> {} by A2,A4,A8,XBOOLE_0:def 4;
  then
A10: L meets A;
  A is connected by CONNSP_1:23;
  hence contradiction by A5,A6,A8,A9,A7,A10,JORDAN2C:92,SPRECT_4:6;
end;
