reserve GS for GraphStruct;
reserve G,G1,G2,G3 for _Graph;
reserve e,x,x1,x2,y,y1,y2,E,V,X,Y for set;
reserve n,n1,n2 for Nat;
reserve v,v1,v2 for Vertex of G;

theorem
  for G being loopless _Graph, v being Vertex of G holds
    v.degree() = card v.edgesInOut()
proof
  let G be loopless _Graph, v be Vertex of G;
  set In = v.edgesIn(), Out = v.edgesOut();
  now
    given e being object such that
A1: e in In /\ Out;
    e in Out by A1,XBOOLE_0:def 4;
    then
A2: (the_Source_of G).e = v by Lm8;
    e in In by A1,XBOOLE_0:def 4;
    then (the_Target_of G).e = v by Lm7;
    hence contradiction by A1,A2,Def18;
  end;
  then In /\ Out = {} by XBOOLE_0:def 1;
  then In misses Out by XBOOLE_0:def 7;
  hence thesis by CARD_2:35;
end;
