let C be Simple_closed_curve; :: thesis: for n being Element of NAT st n is_sufficiently_large_for C holds
C misses RightComp (Span C,n)
let n be Element of NAT ; :: thesis: ( n is_sufficiently_large_for C implies C misses RightComp (Span C,n) )
set f = Span C,n;
RightComp (Span C,n) is_a_component_of (L~ (Span C,n)) ` by GOBOARD9:def 2;
then A1: ex L being Subset of ((TOP-REAL 2) | ((L~ (Span C,n)) ` )) st
( L = RightComp (Span C,n) & L is_a_component_of (TOP-REAL 2) | ((L~ (Span C,n)) ` ) ) by CONNSP_1:def 6;
LeftComp (Span C,n) is_a_component_of (L~ (Span C,n)) ` by GOBOARD9:def 1;
then A2: ex R being Subset of ((TOP-REAL 2) | ((L~ (Span C,n)) ` )) st
( R = LeftComp (Span C,n) & R is_a_component_of (TOP-REAL 2) | ((L~ (Span C,n)) ` ) ) by CONNSP_1:def 6;
assume A3: n is_sufficiently_large_for C ; :: thesis: C misses RightComp (Span C,n)
then A4: C misses L~ (Span C,n) by Th9;
C c= the carrier of ((TOP-REAL 2) | ((L~ (Span 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~ (Span C,n)) ` )) )
assume A5: c in C ; :: thesis: c in the carrier of ((TOP-REAL 2) | ((L~ (Span C,n)) ` ))
then not c in L~ (Span C,n) by A4, XBOOLE_0:3;
then c in (L~ (Span C,n)) ` by A5, SUBSET_1:50;
hence c in the carrier of ((TOP-REAL 2) | ((L~ (Span C,n)) ` )) by PRE_TOPC:29; :: thesis: verum

end;

then reconsider C1 = C as Subset of ((TOP-REAL 2) | ((L~ (Span C,n)) ` )) ;
assume A6: C meets RightComp (Span C,n) ; :: thesis: contradiction
A7: C1 is connected by CONNSP_1:24;
C meets LeftComp (Span C,n) by A3, Th10;
hence contradiction by A6, A1, A2, A7, JORDAN2C:100, SPRECT_4:7; :: thesis: verum