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

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

end;

hence G .minOutDegree() is zero by Th38; :: thesis: verum