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 real-elabeled

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

end;

suppose not v in the_Vertices_of G ; :: thesis: G .labelVertex (v,x) is real-elabeled

hence G .labelVertex (v,x) is real-elabeled by Def22; :: thesis: verum

end;

end;

end;

hence G .labelVertex (v,x) is real-elabeled ; :: thesis: verum