set q = the empty oriented Chain of G;
set x = the Element of G;
reconsider p = <* the Element of G*> as FinSequence of the carrier of G ;
A1: p is_oriented_vertex_seq_of the empty oriented Chain of G by Lm1;
for n, m being Nat st 1 <= n & n < m & m <= len p & p . n = p . m holds
( n = 1 & m = len p )

proof

let n, m be Nat; :: thesis: ( 1 <= n & n < m & m <= len p & p . n = p . m implies ( n = 1 & m = len p ) )
assume that
A2: 1 <= n and
A3: n < m and
A4: m <= len p and
p . n = p . m ; :: thesis: ( n = 1 & m = len p )
1 < m by A2, A3, XXREAL_0:2;
hence ( n = 1 & m = len p ) by A4, FINSEQ_1:39; :: thesis: verum

end;

then the empty oriented Chain of G is Simple by A1;
hence ex b₁ being oriented Chain of G st b₁ is Simple ; :: thesis: verum