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 e,x,y being object holds
  e in v.edgesInOut() & e Joins x,y,G implies v = x or v = y
proof let e,x,y be object;
  assume that
A1: e in v.edgesInOut() and
A2: e Joins x,y,G;
  now
    assume
A3: v <> x;
    now
      per cases by A1,Th61;
      suppose
        (the_Source_of G).e = v;
        hence v = y by A2,A3;
      end;
      suppose
        (the_Target_of G).e = v;
        hence v = y by A2,A3;
      end;
    end;
    hence v = y;
  end;
  hence thesis;
end;
