let G1 be _Graph; :: thesis: for G2 being Subgraph of G1
for W1 being Walk of G1
for W2 being Walk of G2 st W1 = W2 holds
( W1 is Cycle-like iff W2 is Cycle-like )
let G2 be Subgraph of G1; :: thesis: for W1 being Walk of G1
for W2 being Walk of G2 st W1 = W2 holds
( W1 is Cycle-like iff W2 is Cycle-like )
let W1 be Walk of G1; :: thesis: for W2 being Walk of G2 st W1 = W2 holds
( W1 is Cycle-like iff W2 is Cycle-like )
let W2 be Walk of G2; :: thesis: ( W1 = W2 implies ( W1 is Cycle-like iff W2 is Cycle-like ) )
assume A1: W1 = W2 ; :: thesis: ( W1 is Cycle-like iff W2 is Cycle-like )

hereby :: thesis: ( W2 is Cycle-like implies W1 is Cycle-like )

assume W1 is Cycle-like ; :: thesis: W2 is Cycle-like
then ( W2 is closed & W2 is Path-like & W2 is trivial ) by A1, GLIB_001:176;
hence W2 is Cycle-like ; :: thesis: verum

end;

assume W2 is Cycle-like ; :: thesis: W1 is Cycle-like
then ( W1 is closed & W1 is Path-like & W1 is trivial ) by A1, GLIB_001:176;
hence W1 is Cycle-like ; :: thesis: verum