
theorem Th41:
  for G2 being _Graph, V being set, G1 being addVertices of G2, V
  for x,y being set, e being object holds
    (e Joins x,y,G1 iff e Joins x,y,G2) &
    (e DJoins x,y,G1 iff e DJoins x,y,G2) &
    (e SJoins x,y,G1 iff e SJoins x,y,G2) &
    (e DSJoins x,y,G1 iff e DSJoins x,y,G2)
proof
  let G2 be _Graph, V be set, G1 be addVertices of G2, V;
  let x,y be set, e be object;
  set v1 = (the_Source_of G1).e, w1 = (the_Target_of G1).e;
  set v2 = (the_Source_of G2).e, w2 = (the_Target_of G2).e;
  hereby
    assume e Joins x,y,G1;
    then e in the_Edges_of G1 & (v1 = x & w1 = y or v1 = y & w1 = x)
      by GLIB_000:def 13;
    then A1: e in the_Edges_of G2 & (v1 = x & w1 = y or v1 = y & w1 = x)
      by GLIB_006:def 10;
    then v1 = v2 & w1 = w2 by GLIB_006:def 9;
    hence e Joins x,y,G2 by A1, GLIB_000:def 13;
  end;
  thus e Joins x,y,G2 implies e Joins x,y,G1 by GLIB_006:70;
  hereby
    assume e DJoins x,y,G1;
    then e in the_Edges_of G1 & v1 = x & w1 = y by GLIB_000:def 14;
    then A2: e in the_Edges_of G2 & v1 = x & w1 = y by GLIB_006:def 10;
    then v1 = v2 & w1 = w2 by GLIB_006:def 9;
    hence e DJoins x,y,G2 by A2, GLIB_000:def 14;
  end;
  thus e DJoins x,y,G2 implies e DJoins x,y,G1 by GLIB_006:70;
  hereby
    assume e SJoins x,y,G1;
    then e in the_Edges_of G1 & (v1 in x & w1 in y or v1 in y & w1 in x)
      by GLIB_000:def 15;
    then A3: e in the_Edges_of G2 & (v1 in x & w1 in y or v1 in y & w1 in x)
      by GLIB_006:def 10;
    then v1 = v2 & w1 = w2 by GLIB_006:def 9;
    hence e SJoins x,y,G2 by A3, GLIB_000:def 15;
  end;
  thus e SJoins x,y,G2 implies e SJoins x,y,G1 by GLIB_006:70;
  hereby
    assume e DSJoins x,y,G1;
    then e in the_Edges_of G1 & v1 in x & w1 in y by GLIB_000:def 16;
    then A4: e in the_Edges_of G2 & v1 in x & w1 in y by GLIB_006:def 10;
    then v1 = v2 & w1 = w2 by GLIB_006:def 9;
    hence e DSJoins x,y,G2 by A4, GLIB_000:def 16;
  end;
  thus e DSJoins x,y,G2 implies e DSJoins x,y,G1 by GLIB_006:70;
end;
