let G be Graph; :: thesis: for IT being oriented Chain of G
for vs being FinSequence of the carrier of G st IT is Simple & vs is_oriented_vertex_seq_of IT holds
for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs )
let IT be oriented Chain of G; :: thesis: for vs being FinSequence of the carrier of G st IT is Simple & vs is_oriented_vertex_seq_of IT holds
for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs )
let vs be FinSequence of the carrier of G; :: thesis: ( IT is Simple & vs is_oriented_vertex_seq_of IT implies for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs ) )
assume that
A1: IT is Simple and
A2: vs is_oriented_vertex_seq_of IT ; :: thesis: for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs )
A3: len vs = (len IT) + 1 by A2, GRAPH_4:def 5;
consider vs' being FinSequence of the carrier of G such that
A4: vs' is_oriented_vertex_seq_of IT and
A5: for n, m being Element of NAT st 1 <= n & n < m & m <= len vs' & vs' . n = vs' . m holds
( n = 1 & m = len vs' ) by A1, GRAPH_4:def 7;

per cases ( IT <> {} or IT = {} ) ;

suppose IT <> {} ; :: thesis: for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs )

then vs = vs' by A2, A4, GRAPH_4:11;
hence for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs ) by A5; :: thesis: verum

end;

suppose IT = {} ; :: thesis: for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs )

then len IT = 0 ;
hence for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs ) by A3, XXREAL_0:2; :: thesis: verum

end;

end;