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 Th148:
  for G being _trivial _Graph, v being Vertex of G holds
    v.edgesIn() = the_Edges_of G & v.edgesOut() = the_Edges_of G &
    v.edgesInOut() = the_Edges_of G
proof
  let G be _trivial _Graph, v be Vertex of G;
  consider v0 being Vertex of G such that
    A1: the_Vertices_of G = {v0} by Th22;
  A2: v = v0 by A1, TARSKI:def 1;
  A3: now
    let e be object;
    assume A4: e in the_Edges_of G;
    then e Joins (the_Source_of G).e,(the_Target_of G).e,G;
    then (the_Source_of G).e in the_Vertices_of G &
      (the_Target_of G).e in the_Vertices_of G by FUNCT_2:5;
    then (the_Source_of G).e = v0 & (the_Target_of G).e = v0
      by A1, TARSKI:def 1;
    hence e DJoins v,v,G by A2, A4;
  end;
  for e being object st e in the_Edges_of G holds e in v.edgesIn()
    by A3, Th57;
  then the_Edges_of G c= v.edgesIn();
  hence A5: v.edgesIn() = the_Edges_of G by XBOOLE_0:def 10;
  for e being object st e in the_Edges_of G holds e in v.edgesOut()
    by A3, Th59;
  then the_Edges_of G c= v.edgesOut();
  hence v.edgesOut() = the_Edges_of G by XBOOLE_0:def 10;
  hence v.edgesInOut() = the_Edges_of G by A5;
end;
