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 Th70:
  x in v.outNeighbors() iff ex e being object st e DJoins v,x,G
proof
  hereby
    assume x in v.outNeighbors();
    then consider e being object such that
A1: e in dom (the_Target_of G) and
A2: e in v.edgesOut() and
A3: x = (the_Target_of G).e by FUNCT_1:def 6;
    take e;
    (the_Source_of G).e = v by A2,Lm8;
    hence e DJoins v,x,G by A1,A3;
  end;
  given e being object such that
A4: e DJoins v,x,G;
A5: e in the_Edges_of G by A4;
  then
A6: e in dom (the_Target_of G) by FUNCT_2:def 1;
  (the_Source_of G).e = v by A4;
  then
A7: e in v.edgesOut() by A5,Lm8;
  (the_Target_of G).e = x by A4;
  hence thesis by A7,A6,FUNCT_1:def 6;
end;
