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 Th144:
  for G being _Graph, v being Vertex of G, e,w being object
  st v is isolated holds not e DJoins v,w,G & not e DJoins w,v,G
proof
  let G be _Graph, v be Vertex of G, e,w be object;
  assume A1: v is isolated;
  assume e DJoins v,w,G or e DJoins w,v,G;
  then per cases;
  suppose e DJoins v,w,G;
    then e Joins v,w,G;
    hence contradiction by A1, Th143;
  end;
  suppose e DJoins w,v,G;
    then e Joins v,w,G;
    hence contradiction by A1, Th143;
  end;
end;
