set VE = () \/ ();
hereby :: thesis: ( not e Joins x,y,G implies G .walkOf the Element of the_Vertices_of G is Walk of G )
set W = <*x,e,y*>;
assume A1: e Joins x,y,G ; :: thesis: <*x,e,y*> is Walk of G
then y is Vertex of G by GLIB_000:13;
then A2: y is Element of () \/ () by XBOOLE_0:def 3;
e in the_Edges_of G by ;
then A3: e is Element of () \/ () by XBOOLE_0:def 3;
x is Vertex of G by ;
then x is Element of () \/ () by XBOOLE_0:def 3;
then reconsider W = <*x,e,y*> as FinSequence of () \/ () by ;
A4: W . 1 = x by FINSEQ_1:45;
A5: W . 2 = e by FINSEQ_1:45;
now :: thesis: ( len W is odd & W . 1 in the_Vertices_of G & ( for n being odd Element of NAT st n < len W holds
W . (n + 1) Joins W . n,W . (n + 2),G ) )
reconsider aa1 = 1 as odd Element of NAT by JORDAN12:2;
aa1 + 2 is odd ;
hence len W is odd by FINSEQ_1:45; :: thesis: ( W . 1 in the_Vertices_of G & ( for n being odd Element of NAT st n < len W holds
W . (n + 1) Joins W . n,W . (n + 2),G ) )

thus W . 1 in the_Vertices_of G by ; :: thesis: for n being odd Element of NAT st n < len W holds
W . (n + 1) Joins W . n,W . (n + 2),G

let n be odd Element of NAT ; :: thesis: ( n < len W implies W . (n + 1) Joins W . n,W . (n + 2),G )
assume n < len W ; :: thesis: W . (n + 1) Joins W . n,W . (n + 2),G
then n < 2 + 1 by FINSEQ_1:45;
then n <= 2 by NAT_1:13;
then not not n = 0 & ... & not n = 2 ;
hence W . (n + 1) Joins W . n,W . (n + 2),G by ; :: thesis: verum
end;
hence <*x,e,y*> is Walk of G by Def3; :: thesis: verum
end;
thus ( not e Joins x,y,G implies G .walkOf the Element of the_Vertices_of G is Walk of G ) ; :: thesis: verum