let n be Element of NAT ; :: thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds C misses LeftComp (Cage C,n)
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: C misses LeftComp (Cage C,n)
set f = Cage C,n;
assume A1: C meets LeftComp (Cage C,n) ; :: thesis: contradiction
RightComp (Cage C,n) is_a_component_of (L~ (Cage C,n)) ` by GOBOARD9:def 2;
then A2: ex R being Subset of ((TOP-REAL 2) | ((L~ (Cage C,n)) ` )) st
( R = RightComp (Cage C,n) & R is_a_component_of (TOP-REAL 2) | ((L~ (Cage C,n)) ` ) ) by CONNSP_1:def 6;
C misses L~ (Cage C,n) by Th5;
then A3: C /\ (L~ (Cage C,n)) = {} by XBOOLE_0:def 7;
C c= the carrier of ((TOP-REAL 2) | ((L~ (Cage C,n)) ` ))

proof

let c be set ; :: according to TARSKI:def 3 :: thesis: ( not c in C or c in the carrier of ((TOP-REAL 2) | ((L~ (Cage C,n)) ` )) )
assume A4: c in C ; :: thesis: c in the carrier of ((TOP-REAL 2) | ((L~ (Cage C,n)) ` ))
then not c in L~ (Cage C,n) by A3, XBOOLE_0:def 4;
then c in (L~ (Cage C,n)) ` by A4, SUBSET_1:50;
hence c in the carrier of ((TOP-REAL 2) | ((L~ (Cage C,n)) ` )) by PRE_TOPC:29; :: thesis: verum

end;

then reconsider C1 = C as Subset of ((TOP-REAL 2) | ((L~ (Cage C,n)) ` )) ;
LeftComp (Cage C,n) is_a_component_of (L~ (Cage C,n)) ` by GOBOARD9:def 1;
then A5: ex L being Subset of ((TOP-REAL 2) | ((L~ (Cage C,n)) ` )) st
( L = LeftComp (Cage C,n) & L is_a_component_of (TOP-REAL 2) | ((L~ (Cage C,n)) ` ) ) by CONNSP_1:def 6;
( C meets RightComp (Cage C,n) & C1 is connected ) by Th9, CONNSP_1:24;
hence contradiction by A1, A5, A2, JORDAN2C:100, SPRECT_4:7; :: thesis: verum