let G be locally-finite with_max_out_degree _Graph; :: thesis: for n being Nat holds
( G .maxOutDegree() = n iff ex v being Vertex of G st
( v .outDegree() = n & ( for w being Vertex of G holds w .outDegree() <= v .outDegree() ) ) )
let n be Nat; :: thesis: ( G .maxOutDegree() = n iff ex v being Vertex of G st
( v .outDegree() = n & ( for w being Vertex of G holds w .outDegree() <= v .outDegree() ) ) )

hereby :: thesis: ( ex v being Vertex of G st
( v .outDegree() = n & ( for w being Vertex of G holds w .outDegree() <= v .outDegree() ) ) implies G .maxOutDegree() = n )

assume G .maxOutDegree() = n ; :: thesis: ex v being Vertex of G st
( v .outDegree() = n & ( for w being Vertex of G holds w .outDegree() <= v .outDegree() ) )
then consider v being Vertex of G such that
A1: v .outDegree() = n and
A2: for w being Vertex of G holds w .outDegree() c= v .outDegree() by Th81;
take v = v; :: thesis: ( v .outDegree() = n & ( for w being Vertex of G holds w .outDegree() <= v .outDegree() ) )
thus v .outDegree() = n by A1; :: thesis: for w being Vertex of G holds w .outDegree() <= v .outDegree()
let w be Vertex of G; :: thesis: w .outDegree() <= v .outDegree()
Segm (w .outDegree()) c= Segm (v .outDegree()) by A2;
hence w .outDegree() <= v .outDegree() by NAT_1:39; :: thesis: verum

end;

given v being Vertex of G such that A3: v .outDegree() = n and
A4: for w being Vertex of G holds w .outDegree() <= v .outDegree() ; :: thesis: G .maxOutDegree() = n

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

let w be Vertex of G; :: thesis: w .outDegree() c= v .outDegree()
Segm (w .outDegree()) c= Segm (v .outDegree()) by A4, NAT_1:39;
hence w .outDegree() c= v .outDegree() ; :: thesis: verum

end;

hence G .maxOutDegree() = n by A3, Th50; :: thesis: verum