let G be _Graph; :: thesis: for W1, W3 being Walk of G
for e, u, v being object holds
( W1 is_Walk_from u,v iff W1 .replaceEdgeWith (e,W3) is_Walk_from u,v )
let W1, W3 be Walk of G; :: thesis: for e, u, v being object holds
( W1 is_Walk_from u,v iff W1 .replaceEdgeWith (e,W3) is_Walk_from u,v )
let e, u, v be object ; :: thesis: ( W1 is_Walk_from u,v iff W1 .replaceEdgeWith (e,W3) is_Walk_from u,v )

per cases ( ( e Joins W3 .first() ,W3 .last() ,G & G .walkOf ((W3 .first()),e,(W3 .last())) is_odd_substring_of W1, 0 ) or not e Joins W3 .first() ,W3 .last() ,G or not G .walkOf ((W3 .first()),e,(W3 .last())) is_odd_substring_of W1, 0 ) ;

suppose ( e Joins W3 .first() ,W3 .last() ,G & G .walkOf ((W3 .first()),e,(W3 .last())) is_odd_substring_of W1, 0 ) ; :: thesis: ( W1 is_Walk_from u,v iff W1 .replaceEdgeWith (e,W3) is_Walk_from u,v )

then consider W2 being Walk of G such that
A1: W1 .replaceEdgeWith (e,W3) = W1 .replaceWith (W2,W3) by Th46;
thus ( W1 is_Walk_from u,v iff W1 .replaceEdgeWith (e,W3) is_Walk_from u,v ) by A1, Th50; :: thesis: verum

end;

suppose ( not e Joins W3 .first() ,W3 .last() ,G or not G .walkOf ((W3 .first()),e,(W3 .last())) is_odd_substring_of W1, 0 ) ; :: thesis: ( W1 is_Walk_from u,v iff W1 .replaceEdgeWith (e,W3) is_Walk_from u,v )

hence ( W1 is_Walk_from u,v iff W1 .replaceEdgeWith (e,W3) is_Walk_from u,v ) by Def6; :: thesis: verum

end;

end;