reserve i,j,k,n,m for Nat;
reserve p,q for Point of TOP-REAL 2;
reserve G for Go-board;
reserve C for Subset of TOP-REAL 2;

theorem Th33:
  for f being rectangular special_circular_sequence, g being
  S-Sequence_in_R2 st g/.1 in LeftComp f & g/.len g in RightComp f holds L~f
  meets L~g
proof
  let f be rectangular special_circular_sequence, g be S-Sequence_in_R2 such
  that
A1: g/.1 in LeftComp f and
A2: g/.len g in RightComp f;
A3: len g >= 2 by TOPREAL1:def 8;
  then g/.1 in L~g by JORDAN3:1;
  then g/.1 in LeftComp f /\ L~g by A1,XBOOLE_0:def 4;
  then
A4: LeftComp f meets L~g;
  g/.len g in L~g by A3,JORDAN3:1;
  then g/.len g in RightComp f /\ L~g by A2,XBOOLE_0:def 4;
  then
A5: RightComp f meets L~g;
A6: LeftComp f misses RightComp f by SPRECT_1:88;
  assume L~f misses L~g;
  then L~g c= (L~f)` by SUBSET_1:23;
  then
A7: L~g c= LeftComp f \/ RightComp f by GOBRD12:10;
A8: L~g is_an_arc_of g/.1,g/.len g by TOPREAL1:25;
A9: RightComp f is open by Th24;
  LeftComp f is open by Th24;
  hence contradiction by A7,A9,A4,A5,A6,A8,JORDAN6:10,TOPREAL5:1;
end;
