reserve i,j,k,k1,k2,i1,i2,j1,j2 for Nat,
  r,s for Real,
  x for set,
  f for non constant standard special_circular_sequence;

theorem Th5:
  Down(LeftComp f,(L~f)`) \/ Down(RightComp f,(L~f)`) is
  a_union_of_components of (TOP-REAL 2)|((L~f)`) & Down(LeftComp f,(L~f)`)=
  LeftComp f & Down(RightComp f,(L~f)`)=RightComp f
proof
  LeftComp f is_a_component_of (L~f)` by GOBOARD9:def 1;
  then consider B1 being Subset of (TOP-REAL 2)|((L~f)`) such that
A1: B1 = LeftComp f and
A2: B1 is a_component by CONNSP_1:def 6;
  RightComp f is_a_component_of (L~f)` by GOBOARD9:def 2;
  then consider B2 being Subset of (TOP-REAL 2)|((L~f)`) such that
A3: B2 = RightComp f and
A4: B2 is a_component by CONNSP_1:def 6;
A5: B2 is Subset of (L~f)` by Lm1;
  then
A6: Down(RightComp f,(L~f)`) is a_component by A3,A4,XBOOLE_1:28;
A7: B1 is Subset of (L~f)` by Lm1;
  then
  Down(LeftComp f,(L~f)`) is a_component by A1,A2,XBOOLE_1:28;
  hence thesis by A1,A7,A3,A5,A6,GOBRD11:3,XBOOLE_1:28;
end;
