reserve G for Graph,
  v, v1, v2 for Vertex of G,
  c for Chain of G,
  p, p1, p2 for Path of G,
  vs, vs1, vs2 for FinSequence of the carrier of G,
  e, X for set,
  n, m for Nat;
reserve G for finite Graph,
  v for Vertex of G,
  c for Chain of G,
  vs for FinSequence of the carrier of G,
  X1, X2 for set;

theorem Th22:
  card Edges_In v = EdgesIn v
proof
  consider X being finite set such that
A1: for z being set holds z in X iff z in the carrier' of G & (the
  Target of G).z = v and
A2: EdgesIn v = card(X) by GRAPH_1:def 21;
  now
    let e be object;
    e in Edges_In (v, the carrier' of G) iff e in the carrier' of G & (the
    Target of G).e = v by Def1;
    hence e in Edges_In v iff e in X by A1;
  end;
  hence thesis by A2,TARSKI:2;
end;
