reserve i, j, n for Nat,
  f for non constant standard special_circular_sequence,
  g for clockwise_oriented non constant standard special_circular_sequence,
  p, q for Point of TOP-REAL 2,
  P for Subset of TOP-REAL 2,
  C for compact non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board;

theorem
  Cl LeftComp f = (LeftComp f) \/ L~f
proof
  thus Cl LeftComp f c= LeftComp f \/ L~f
  proof
    let x be object;
    assume
A1: x in Cl LeftComp f;
    now
A2:   not x in RightComp f by A1,Th14,TOPS_1:12;
      assume not x in LeftComp f;
      hence x in L~f by A1,A2,Th16;
    end;
    hence thesis by XBOOLE_0:def 3;
  end;
  (Cl LeftComp f) \ LeftComp f c= Cl LeftComp f by XBOOLE_1:36;
  then LeftComp f c= Cl LeftComp f & L~f c= Cl LeftComp f by Th20,PRE_TOPC:18;
  hence thesis by XBOOLE_1:8;
end;
