let G be _Graph; G .minOutDegree() c= G .minDegree()
consider v1 being Vertex of G such that
A1:
v1 .outDegree() = G .minOutDegree()
and
A2:
for w being Vertex of G holds v1 .outDegree() c= w .outDegree()
by Th38;
consider v2 being Vertex of G such that
A3:
v2 .degree() = G .minDegree()
and
for w being Vertex of G holds v2 .degree() c= w .degree()
by Th36;
A4:
v2 .outDegree() c= G .minDegree()
by A3, CARD_2:94;
v1 .outDegree() c= v2 .outDegree()
by A2;
hence
G .minOutDegree() c= G .minDegree()
by A1, A4, XBOOLE_1:1; verum