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 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 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 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 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 vs9 being FinSequence of the carrier of G such that
A4: vs9 is_oriented_vertex_seq_of IT and
A5: for n, m being Nat st 1 <= n & n < m & m <= len vs9 & vs9 . n = vs9 . m holds
( n = 1 & m = len vs9 ) by A1, GRAPH_4:def 7;
A6: for n, m being Nat st 1 <= n & n < m & m <= len vs9 & vs9 . n = vs9 . m holds
( n = 1 & m = len vs9 ) by A5;
per cases ( IT <> {} or IT = {} ) ;
suppose IT <> {} ; :: thesis: for n, m being Nat st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs )
then vs = vs9 by A2, A4, GRAPH_4:10;
hence for n, m being Nat st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs ) by A6; :: thesis: verum
end;
suppose IT = {} ; :: thesis: for n, m being 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 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;