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;
C misses L~ (Cage C,n) by Th5;
then A1: C /\ (L~ (Cage C,n)) = {} by XBOOLE_0:def 7;
A2: C meets RightComp (Cage C,n) by Th9;
assume A3: C meets LeftComp (Cage C,n) ; :: thesis: contradiction
LeftComp (Cage C,n) is_a_component_of (L~ (Cage C,n)) ` by GOBOARD9:def 1;
then consider L being Subset of ((TOP-REAL 2) | ((L~ (Cage C,n)) ` )) such that
A4: L = LeftComp (Cage C,n) and
A5: L is_a_component_of (TOP-REAL 2) | ((L~ (Cage C,n)) ` ) by CONNSP_1:def 6;
RightComp (Cage C,n) is_a_component_of (L~ (Cage C,n)) ` by GOBOARD9:def 2;
then consider R being Subset of ((TOP-REAL 2) | ((L~ (Cage C,n)) ` )) such that
A6: R = RightComp (Cage C,n) and
A7: R is_a_component_of (TOP-REAL 2) | ((L~ (Cage C,n)) ` ) by CONNSP_1:def 6;
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 c in C ; :: thesis: c in the carrier of ((TOP-REAL 2) | ((L~ (Cage C,n)) ` ))
then ( c in the carrier of (TOP-REAL 2) & not c in L~ (Cage C,n) ) by A1, XBOOLE_0:def 4;
then c in (L~ (Cage C,n)) ` by 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)) ` )) ;
C1 is connected by CONNSP_1:24;
hence contradiction by A2, A3, A4, A5, A6, A7, JORDAN2C:100, SPRECT_4:7; :: thesis: verum