now :: thesis: ex v being Vertex of G st
( v .degree() = 0 & ( for w being Vertex of G holds v .degree() c= w .degree() ) )

take v = the Vertex of G; :: thesis: ( v .degree() = 0 & ( for w being Vertex of G holds v .degree() c= w .degree() ) )
thus v .degree() = 0 ; :: thesis: for w being Vertex of G holds v .degree() c= w .degree()
let w be Vertex of G; :: thesis: v .degree() c= w .degree()
thus v .degree() c= w .degree() ; :: thesis: verum

end;

hence G .minDegree() is zero by Th36; :: thesis: verum