let G1, G2 be _Graph; :: thesis: for x, y being set
for W1 being Walk of G1
for W2 being Walk of G2 st W1 = W2 holds
( W1 is_Walk_from x,y iff W2 is_Walk_from x,y )
let x, y be set ; :: thesis: for W1 being Walk of G1
for W2 being Walk of G2 st W1 = W2 holds
( W1 is_Walk_from x,y iff W2 is_Walk_from x,y )
let W1 be Walk of G1; :: thesis: for W2 being Walk of G2 st W1 = W2 holds
( W1 is_Walk_from x,y iff W2 is_Walk_from x,y )
let W2 be Walk of G2; :: thesis: ( W1 = W2 implies ( W1 is_Walk_from x,y iff W2 is_Walk_from x,y ) )
assume A1: W1 = W2 ; :: thesis: ( W1 is_Walk_from x,y iff W2 is_Walk_from x,y )
( W1 is_Walk_from x,y iff ( W1 .first() = x & W1 .last() = y ) ) by Def23;
then ( W1 is_Walk_from x,y iff ( W2 .first() = x & W2 .last() = y ) ) by A1;
hence ( W1 is_Walk_from x,y iff W2 is_Walk_from x,y ) by Def23; :: thesis: verum