now

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

suppose e in the_Edges_of G ; :: thesis: G .labelEdge e,x is finite

then G .labelEdge e,x = G .set ELabelSelector ,((the_ELabel_of G) +* (e .--> x)) by Def21;
hence G .labelEdge e,x is finite ; :: thesis: verum

end;

suppose not e in the_Edges_of G ; :: thesis: G .labelEdge e,x is finite

hence G .labelEdge e,x is finite by Def21; :: thesis: verum

end;

end;

end;

hence G .labelEdge e,x is finite ; :: thesis: verum