theorem Th64:
  e in v1.edgesInOut() iff ex v2 being Vertex of G st e Joins v1,v2,G
proof
  hereby
    assume
A1: e in v1.edgesInOut();
    now
      per cases by A1,Th61;
      suppose
A2:     (the_Source_of G).e = v1;
        set v2 = (the_Target_of G).e;
        reconsider v2 as Vertex of G by A1,FUNCT_2:5;
        take v2;
        thus e Joins v1, v2, G by A1,A2;
      end;
      suppose
A3:     (the_Target_of G).e = v1;
        set v2 = (the_Source_of G).e;
        reconsider v2 as Vertex of G by A1,FUNCT_2:5;
        take v2;
        thus e Joins v1,v2,G by A1,A3;
      end;
    end;
    hence ex v2 being Vertex of G st e Joins v1,v2,G;
  end;
  given v2 being Vertex of G such that
A4: e Joins v1,v2,G;
  thus thesis by A4,Th62;
end;
