let G1 be _Graph; :: thesis: for G2 being Subgraph of G1
for W1 being Walk of G1
for W2 being Walk of G2
for e being set st W1 = W2 & e in (W2 .last()) .edgesInOut() holds
W1 .addEdge e = W2 .addEdge e
let G2 be Subgraph of G1; :: thesis: for W1 being Walk of G1
for W2 being Walk of G2
for e being set st W1 = W2 & e in (W2 .last()) .edgesInOut() holds
W1 .addEdge e = W2 .addEdge e
let W1 be Walk of G1; :: thesis: for W2 being Walk of G2
for e being set st W1 = W2 & e in (W2 .last()) .edgesInOut() holds
W1 .addEdge e = W2 .addEdge e
let W2 be Walk of G2; :: thesis: for e being set st W1 = W2 & e in (W2 .last()) .edgesInOut() holds
W1 .addEdge e = W2 .addEdge e
let e be set ; :: thesis: ( W1 = W2 & e in (W2 .last()) .edgesInOut() implies W1 .addEdge e = W2 .addEdge e )
assume that
A1: W1 = W2 and
A2: e in (W2 .last()) .edgesInOut() ; :: thesis: W1 .addEdge e = W2 .addEdge e
set W2B = G2 .walkOf ((W2 .last()),e,((W2 .last()) .adj e));
set W1B = G1 .walkOf ((W1 .last()),e,((W1 .last()) .adj e));
A3: e Joins W2 .last() ,(W2 .last()) .adj e,G2 by A2, GLIB_000:67;
(W1 .last()) .adj e = (W2 .last()) .adj e by A1, A2, GLIB_000:80;
then G1 .walkOf ((W1 .last()),e,((W1 .last()) .adj e)) = G2 .walkOf ((W2 .last()),e,((W2 .last()) .adj e)) by A1, A3, Th171;
hence W1 .addEdge e = W2 .addEdge e by A1, Th33; :: thesis: verum