let G be _Graph; :: thesis: for W1, W2 being Walk of G
for e, u, v being object holds
( W1 is_Walk_from u,v iff W1 .replaceWithEdge (W2,e) is_Walk_from u,v )
let W1, W2 be Walk of G; :: thesis: for e, u, v being object holds
( W1 is_Walk_from u,v iff W1 .replaceWithEdge (W2,e) is_Walk_from u,v )
let e, u, v be object ; :: thesis: ( W1 is_Walk_from u,v iff W1 .replaceWithEdge (W2,e) is_Walk_from u,v )

per cases ( ( W2 is_odd_substring_of W1, 0 & e Joins W2 .first() ,W2 .last() ,G ) or not W2 is_odd_substring_of W1, 0 or not e Joins W2 .first() ,W2 .last() ,G ) ;

suppose ( W2 is_odd_substring_of W1, 0 & e Joins W2 .first() ,W2 .last() ,G ) ; :: thesis: ( W1 is_Walk_from u,v iff W1 .replaceWithEdge (W2,e) is_Walk_from u,v )

then consider W3 being Walk of G such that
A1: W1 .replaceWithEdge (W2,e) = W1 .replaceWith (W2,W3) by Th47;
thus ( W1 is_Walk_from u,v iff W1 .replaceWithEdge (W2,e) is_Walk_from u,v ) by A1, Th50; :: thesis: verum

end;

suppose ( not W2 is_odd_substring_of W1, 0 or not e Joins W2 .first() ,W2 .last() ,G ) ; :: thesis: ( W1 is_Walk_from u,v iff W1 .replaceWithEdge (W2,e) is_Walk_from u,v )

hence ( W1 is_Walk_from u,v iff W1 .replaceWithEdge (W2,e) is_Walk_from u,v ) by Def7; :: thesis: verum

end;

end;