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

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

assume G .maxDegree() = n ; :: thesis: ex v being Vertex of G st
( v .degree() = n & ( for w being Vertex of G holds w .degree() <= v .degree() ) )
then consider v being Vertex of G such that
A1: v .degree() = n and
A2: for w being Vertex of G holds w .degree() c= v .degree() by Th79;
take v = v; :: thesis: ( v .degree() = n & ( for w being Vertex of G holds w .degree() <= v .degree() ) )
thus v .degree() = n by A1; :: thesis: for w being Vertex of G holds w .degree() <= v .degree()
let w be Vertex of G; :: thesis: w .degree() <= v .degree()
Segm (w .degree()) c= Segm (v .degree()) by A2;
hence w .degree() <= v .degree() by NAT_1:39; :: thesis: verum

end;

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

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

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

end;

hence G .maxDegree() = n by A3, Th48; :: thesis: verum