consider q being empty oriented Chain of G;
consider x being Element of the carrier of G;
reconsider p = <*x*> as FinSequence of the carrier of G ;
A1: p is_oriented_vertex_seq_of q by Lm1;
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 )

proof

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

end;

then q is Simple by A1, Def7;
hence ex b₁ being oriented Chain of G st b₁ is Simple ; :: thesis: verum