consider q being empty Chain of G;
reconsider q = q as V7() Chain of G ;
take q ; :: thesis: q is simple
consider x being Element of the carrier of G;
reconsider p = <*x*> as FinSequence of the carrier of G ;
take p ; :: according to GRAPH_2:def 10 :: thesis: ( p is_vertex_seq_of q & ( for n, m being Element of NAT st 1 <= n & n < m & m <= len p & p . n = p . m holds
( n = 1 & m = len p ) ) )
thus p is_vertex_seq_of q by Lm8; :: thesis: for n, m being Element of NAT st 1 <= n & n < m & m <= len p & p . n = p . m holds
( n = 1 & m = len p )
let n, m be Element of NAT ; :: thesis: ( 1 <= n & n < m & m <= len p & p . n = p . m implies ( n = 1 & m = len p ) )
assume ( 1 <= n & n < m & m <= len p & p . n = p . m ) ; :: thesis: ( n = 1 & m = len p )
then ( 1 < m & m <= 1 ) by FINSEQ_1:56, XXREAL_0:2;
hence ( n = 1 & m = len p ) ; :: thesis: verum