let G1, G2 be _Graph; :: thesis: for F being PGraphMapping of G1,G2 st F is onto & F is semi-Dcontinuous holds
G2 .supDegree() c= G1 .supDegree()
let F be PGraphMapping of G1,G2; :: thesis: ( F is onto & F is semi-Dcontinuous implies G2 .supDegree() c= G1 .supDegree() )
assume A1: ( F is onto & F is semi-Dcontinuous ) ; :: thesis: G2 .supDegree() c= G1 .supDegree()
set D1 = { (v .degree()) where v is Vertex of G1 : verum } ;
set D2 = { (w .degree()) where w is Vertex of G2 : verum } ;

now :: thesis: for x being object st x in G2 .supDegree() holds
x in G1 .supDegree()

let x be object ; :: thesis: ( x in G2 .supDegree() implies x in G1 .supDegree() )
assume x in G2 .supDegree() ; :: thesis: x in G1 .supDegree()
then consider d2 being set such that
A2: ( x in d2 & d2 in { (w .degree()) where w is Vertex of G2 : verum } ) by TARSKI:def 4;
consider w being Vertex of G2 such that
A3: d2 = w .degree() by A2;
rng (F _V) = the_Vertices_of G2 by A1, GLIB_010:def 12;
then consider v being object such that
A4: ( v in dom (F _V) & (F _V) . v = w ) by FUNCT_1:def 3;
reconsider v = v as Vertex of G1 by A4;
((F _V) /. v) .degree() c= v .degree() by A1, A4, GLIBPRE0:91;
then w .degree() c= v .degree() by A4, PARTFUN1:def 6;
then A5: x in v .degree() by A2, A3;
v .degree() in { (v .degree()) where v is Vertex of G1 : verum } ;
hence x in G1 .supDegree() by A5, TARSKI:def 4; :: thesis: verum

end;

hence G2 .supDegree() c= G1 .supDegree() by TARSKI:def 3; :: thesis: verum