
theorem Th119:
  for S being GraphUnionSet, G being GraphUnion of S
  for e,v,w being object st e DJoins v,w,G
  ex H being Element of S st e DJoins v,w,H
proof
  let S be GraphUnionSet, G be GraphUnion of S;
  let e,v,w be object;
  assume e DJoins v,w,G;
  then A1: e in the_Edges_of G & (the_Source_of G).e = v &
    (the_Target_of G).e = w by GLIB_000:def 14;
  e in union the_Edges_of S by A1, GLIB_014:def 25;
  then consider E being set such that
    A2: e in E & E in the_Edges_of S by TARSKI:def 4;
  consider H being _Graph such that
    A3: H in S & E = the_Edges_of H by A2, GLIB_014:def 15;
  reconsider H as Element of S by A3;
  take H;
  A4: e in dom the_Source_of H & e in dom the_Target_of H
    by A2, A3, FUNCT_2:def 1;
  the_Source_of H in the_Source_of S by GLIB_014:def 16;
  then A5: (the_Source_of H).e = (union the_Source_of S).e by A4, COMPUT_1:13
    .= v by A1, GLIB_014:def 25;
  the_Target_of H in the_Target_of S by GLIB_014:def 17;
  then (the_Target_of H).e = (union the_Target_of S).e by A4, COMPUT_1:13
    .= w by A1, GLIB_014:def 25;
  hence e DJoins v,w,H by A2, A3, A5, GLIB_000:def 14;
end;
