let G be Graph; :: thesis: for c being Chain of G
for vs being FinSequence of the carrier of G st vs is_vertex_seq_of c holds
for n being Nat st 1 <= n & n <= len c holds
( 1 <= n & n <= len vs & 1 <= n + 1 & n + 1 <= len vs & ( ( vs . n = the Target of G . (c . n) & vs . (n + 1) = the Source of G . (c . n) ) or ( vs . n = the Source of G . (c . n) & vs . (n + 1) = the Target of G . (c . n) ) ) )
let c be Chain of G; :: thesis: for vs being FinSequence of the carrier of G st vs is_vertex_seq_of c holds
for n being Nat st 1 <= n & n <= len c holds
( 1 <= n & n <= len vs & 1 <= n + 1 & n + 1 <= len vs & ( ( vs . n = the Target of G . (c . n) & vs . (n + 1) = the Source of G . (c . n) ) or ( vs . n = the Source of G . (c . n) & vs . (n + 1) = the Target of G . (c . n) ) ) )
let vs be FinSequence of the carrier of G; :: thesis: ( vs is_vertex_seq_of c implies for n being Nat st 1 <= n & n <= len c holds
( 1 <= n & n <= len vs & 1 <= n + 1 & n + 1 <= len vs & ( ( vs . n = the Target of G . (c . n) & vs . (n + 1) = the Source of G . (c . n) ) or ( vs . n = the Source of G . (c . n) & vs . (n + 1) = the Target of G . (c . n) ) ) ) )
assume A1: vs is_vertex_seq_of c ; :: thesis: for n being Nat st 1 <= n & n <= len c holds
( 1 <= n & n <= len vs & 1 <= n + 1 & n + 1 <= len vs & ( ( vs . n = the Target of G . (c . n) & vs . (n + 1) = the Source of G . (c . n) ) or ( vs . n = the Source of G . (c . n) & vs . (n + 1) = the Target of G . (c . n) ) ) )
let n be Nat; :: thesis: ( 1 <= n & n <= len c implies ( 1 <= n & n <= len vs & 1 <= n + 1 & n + 1 <= len vs & ( ( vs . n = the Target of G . (c . n) & vs . (n + 1) = the Source of G . (c . n) ) or ( vs . n = the Source of G . (c . n) & vs . (n + 1) = the Target of G . (c . n) ) ) ) )
assume A2: ( 1 <= n & n <= len c ) ; :: thesis: ( 1 <= n & n <= len vs & 1 <= n + 1 & n + 1 <= len vs & ( ( vs . n = the Target of G . (c . n) & vs . (n + 1) = the Source of G . (c . n) ) or ( vs . n = the Source of G . (c . n) & vs . (n + 1) = the Target of G . (c . n) ) ) )
A3: len vs = (len c) + 1 by A1, GRAPH_2:def 7;
thus 1 <= n by A2; :: thesis: ( n <= len vs & 1 <= n + 1 & n + 1 <= len vs & ( ( vs . n = the Target of G . (c . n) & vs . (n + 1) = the Source of G . (c . n) ) or ( vs . n = the Source of G . (c . n) & vs . (n + 1) = the Target of G . (c . n) ) ) )
len c <= (len c) + 1 by NAT_1:11;
hence A4: n <= len vs by A2, A3, XXREAL_0:2; :: thesis: ( 1 <= n + 1 & n + 1 <= len vs & ( ( vs . n = the Target of G . (c . n) & vs . (n + 1) = the Source of G . (c . n) ) or ( vs . n = the Source of G . (c . n) & vs . (n + 1) = the Target of G . (c . n) ) ) )
thus 1 <= n + 1 by NAT_1:11; :: thesis: ( n + 1 <= len vs & ( ( vs . n = the Target of G . (c . n) & vs . (n + 1) = the Source of G . (c . n) ) or ( vs . n = the Source of G . (c . n) & vs . (n + 1) = the Target of G . (c . n) ) ) )
thus A5: n + 1 <= len vs by A2, A3, XREAL_1:8; :: thesis: ( ( vs . n = the Target of G . (c . n) & vs . (n + 1) = the Source of G . (c . n) ) or ( vs . n = the Source of G . (c . n) & vs . (n + 1) = the Target of G . (c . n) ) )
reconsider n = n as Element of NAT by ORDINAL1:def 13;
A6: c . n joins vs /. n,vs /. (n + 1) by A1, A2, GRAPH_2:def 7;
( vs . n = vs /. n & vs . (n + 1) = vs /. (n + 1) ) by A2, A4, A5, FINSEQ_4:24, NAT_1:11;
hence ( ( vs . n = the Target of G . (c . n) & vs . (n + 1) = the Source of G . (c . n) ) or ( vs . n = the Source of G . (c . n) & vs . (n + 1) = the Target of G . (c . n) ) ) by A6, GRAPH_1:def 10; :: thesis: verum