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

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

assume G .minInDegree() = n ; :: thesis: ex v being Vertex of G st
( v .inDegree() = n & ( for w being Vertex of G holds v .inDegree() <= w .inDegree() ) )
then consider v being Vertex of G such that
A1: v .inDegree() = n and
A2: for w being Vertex of G holds v .inDegree() c= w .inDegree() by Th37;
take v = v; :: thesis: ( v .inDegree() = n & ( for w being Vertex of G holds v .inDegree() <= w .inDegree() ) )
thus v .inDegree() = n by A1; :: thesis: for w being Vertex of G holds v .inDegree() <= w .inDegree()
let w be Vertex of G; :: thesis: v .inDegree() <= w .inDegree()
Segm (v .inDegree()) c= Segm (w .inDegree()) by A2;
hence v .inDegree() <= w .inDegree() by NAT_1:39; :: thesis: verum

end;

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

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

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

end;

hence G .minInDegree() = n by A3, Th37; :: thesis: verum