now

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

suppose A1: v in the_Vertices_of G ; :: thesis: G .labelVertex v,x is real-vlabeled

set EL = (the_VLabel_of G) +* (v .--> x);
rng (the_VLabel_of G) c= REAL ;
then the_VLabel_of G is PartFunc of (dom (the_VLabel_of G)),REAL by RELSET_1:11;
then the_VLabel_of G is PartFunc of (the_Vertices_of G),REAL by Lm2, RELSET_1:13;
then reconsider EL = (the_VLabel_of G) +* (v .--> x) as PartFunc of (the_Vertices_of G),REAL by A1, Lm4;
G .labelVertex v,x = G .set VLabelSelector ,EL by A1, Def22;
hence G .labelVertex v,x is real-vlabeled ; :: thesis: verum

end;

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

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

end;

end;

end;

hence G .labelVertex v,x is real-vlabeled ; :: thesis: verum