let G2 be _Graph; :: thesis: for G1 being Supergraph of G2
for W2 being Walk of G2
for W1 being Walk of G1 st W1 = W2 holds
( ( W1 is closed implies W2 is closed ) & ( W2 is closed implies W1 is closed ) & ( W1 is directed implies W2 is directed ) & ( W2 is directed implies W1 is directed ) & ( W1 is trivial implies W2 is trivial ) & ( W2 is trivial implies W1 is trivial ) & ( W1 is Trail-like implies W2 is Trail-like ) & ( W2 is Trail-like implies W1 is Trail-like ) & ( W1 is Path-like implies W2 is Path-like ) & ( W2 is Path-like implies W1 is Path-like ) & ( W1 is vertex-distinct implies W2 is vertex-distinct ) & ( W2 is vertex-distinct implies W1 is vertex-distinct ) & ( W1 is Cycle-like implies W2 is Cycle-like ) & ( W2 is Cycle-like implies W1 is Cycle-like ) )
let G1 be Supergraph of G2; :: thesis: for W2 being Walk of G2
for W1 being Walk of G1 st W1 = W2 holds
( ( W1 is closed implies W2 is closed ) & ( W2 is closed implies W1 is closed ) & ( W1 is directed implies W2 is directed ) & ( W2 is directed implies W1 is directed ) & ( W1 is trivial implies W2 is trivial ) & ( W2 is trivial implies W1 is trivial ) & ( W1 is Trail-like implies W2 is Trail-like ) & ( W2 is Trail-like implies W1 is Trail-like ) & ( W1 is Path-like implies W2 is Path-like ) & ( W2 is Path-like implies W1 is Path-like ) & ( W1 is vertex-distinct implies W2 is vertex-distinct ) & ( W2 is vertex-distinct implies W1 is vertex-distinct ) & ( W1 is Cycle-like implies W2 is Cycle-like ) & ( W2 is Cycle-like implies W1 is Cycle-like ) )
A1: G2 is Subgraph of G1 by Th61;
let W2 be Walk of G2; :: thesis: for W1 being Walk of G1 st W1 = W2 holds
( ( W1 is closed implies W2 is closed ) & ( W2 is closed implies W1 is closed ) & ( W1 is directed implies W2 is directed ) & ( W2 is directed implies W1 is directed ) & ( W1 is trivial implies W2 is trivial ) & ( W2 is trivial implies W1 is trivial ) & ( W1 is Trail-like implies W2 is Trail-like ) & ( W2 is Trail-like implies W1 is Trail-like ) & ( W1 is Path-like implies W2 is Path-like ) & ( W2 is Path-like implies W1 is Path-like ) & ( W1 is vertex-distinct implies W2 is vertex-distinct ) & ( W2 is vertex-distinct implies W1 is vertex-distinct ) & ( W1 is Cycle-like implies W2 is Cycle-like ) & ( W2 is Cycle-like implies W1 is Cycle-like ) )
let W1 be Walk of G1; :: thesis: ( W1 = W2 implies ( ( W1 is closed implies W2 is closed ) & ( W2 is closed implies W1 is closed ) & ( W1 is directed implies W2 is directed ) & ( W2 is directed implies W1 is directed ) & ( W1 is trivial implies W2 is trivial ) & ( W2 is trivial implies W1 is trivial ) & ( W1 is Trail-like implies W2 is Trail-like ) & ( W2 is Trail-like implies W1 is Trail-like ) & ( W1 is Path-like implies W2 is Path-like ) & ( W2 is Path-like implies W1 is Path-like ) & ( W1 is vertex-distinct implies W2 is vertex-distinct ) & ( W2 is vertex-distinct implies W1 is vertex-distinct ) & ( W1 is Cycle-like implies W2 is Cycle-like ) & ( W2 is Cycle-like implies W1 is Cycle-like ) ) )
assume W1 = W2 ; :: thesis: ( ( W1 is closed implies W2 is closed ) & ( W2 is closed implies W1 is closed ) & ( W1 is directed implies W2 is directed ) & ( W2 is directed implies W1 is directed ) & ( W1 is trivial implies W2 is trivial ) & ( W2 is trivial implies W1 is trivial ) & ( W1 is Trail-like implies W2 is Trail-like ) & ( W2 is Trail-like implies W1 is Trail-like ) & ( W1 is Path-like implies W2 is Path-like ) & ( W2 is Path-like implies W1 is Path-like ) & ( W1 is vertex-distinct implies W2 is vertex-distinct ) & ( W2 is vertex-distinct implies W1 is vertex-distinct ) & ( W1 is Cycle-like implies W2 is Cycle-like ) & ( W2 is Cycle-like implies W1 is Cycle-like ) )
hence ( ( W1 is closed implies W2 is closed ) & ( W2 is closed implies W1 is closed ) & ( W1 is directed implies W2 is directed ) & ( W2 is directed implies W1 is directed ) & ( W1 is trivial implies W2 is trivial ) & ( W2 is trivial implies W1 is trivial ) & ( W1 is Trail-like implies W2 is Trail-like ) & ( W2 is Trail-like implies W1 is Trail-like ) & ( W1 is Path-like implies W2 is Path-like ) & ( W2 is Path-like implies W1 is Path-like ) & ( W1 is vertex-distinct implies W2 is vertex-distinct ) & ( W2 is vertex-distinct implies W1 is vertex-distinct ) & ( W1 is Cycle-like implies W2 is Cycle-like ) & ( W2 is Cycle-like implies W1 is Cycle-like ) ) by A1, Th58, GLIB_001:176; :: thesis: verum