now

per cases ( v in the_Vertices_of G or not v in the_Vertices_of G ) ;

suppose v in the_Vertices_of G ; :: thesis: G .labelVertex v,x is finite

then G .labelVertex v,x = G .set VLabelSelector ,((the_VLabel_of G) +* (v .--> x)) by Def22;
hence G .labelVertex v,x is finite ; :: thesis: verum

end;

suppose not v in the_Vertices_of G ; :: thesis: G .labelVertex v,x is finite

hence G .labelVertex v,x is finite by Def22; :: thesis: verum

end;

end;

end;

hence G .labelVertex v,x is finite ; :: thesis: verum