set X = v .--> val;
reconsider X = v .--> val as PartFunc of {v},{val} ;
dom (v .--> val) = {v} ;
then reconsider X = X as PartFunc of (the_Vertices_of G),{val} by RELSET_1:5;
for x being object st x in {val} holds
x in REAL by XREAL_0:def 1;
then {val} c= REAL by TARSKI:def 3;
then reconsider X = X as PartFunc of (the_Vertices_of G),REAL by RELSET_1:7;
G .set (VLabelSelector,X) is real-vlabeled ;
hence G .set (VLabelSelector,(v .--> val)) is real-vlabeled ; :: thesis: verum