let G2 be _Graph; :: thesis: for v1, v2 being Vertex of G2
for e being object
for G1 being addEdge of G2,v1,e,v2
for W being Walk of G1 st ( e in W .edges() implies e in the_Edges_of G2 ) holds
W is Walk of G2
let v1, v2 be Vertex of G2; :: thesis: for e being object
for G1 being addEdge of G2,v1,e,v2
for W being Walk of G1 st ( e in W .edges() implies e in the_Edges_of G2 ) holds
W is Walk of G2
let e be object ; :: thesis: for G1 being addEdge of G2,v1,e,v2
for W being Walk of G1 st ( e in W .edges() implies e in the_Edges_of G2 ) holds
W is Walk of G2
let G1 be addEdge of G2,v1,e,v2; :: thesis: for W being Walk of G1 st ( e in W .edges() implies e in the_Edges_of G2 ) holds
W is Walk of G2
let W be Walk of G1; :: thesis: ( ( e in W .edges() implies e in the_Edges_of G2 ) implies W is Walk of G2 )
assume ( e in W .edges() implies e in the_Edges_of G2 ) ; :: thesis: W is Walk of G2

per cases then ( not e in W .edges() or e in the_Edges_of G2 ) ;

suppose not e in W .edges() ; :: thesis: W is Walk of G2

then A1: W .edges() misses {e} by ZFMISC_1:50;
A2: G2 is Subgraph of G1 by Th61;

per cases ( not e in the_Edges_of G2 or e in the_Edges_of G2 ) ;

suppose A3: not e in the_Edges_of G2 ; :: thesis: W is Walk of G2

W .edges() c= the_Edges_of G1 ;
then W .edges() c= (the_Edges_of G2) \/ {e} by A3, Def11;
then A4: W .edges() c= the_Edges_of G2 by A1, XBOOLE_1:73;
W .vertices() c= the_Vertices_of G1 ;
then W .vertices() c= the_Vertices_of G2 by A3, Def11;
hence W is Walk of G2 by A2, A4, GLIB_001:170; :: thesis: verum

end;

suppose e in the_Edges_of G2 ; :: thesis: W is Walk of G2

then G1 == G2 by Def11;
hence W is Walk of G2 by GLIB_001:179; :: thesis: verum

end;

end;

end;

suppose e in the_Edges_of G2 ; :: thesis: W is Walk of G2

then G1 == G2 by Def11;
hence W is Walk of G2 by GLIB_001:179; :: thesis: verum

end;

end;