
theorem
  for S being GraphUnionSet, G being GraphUnion of S
  for e,v,w being object st e Joins v,w,G
  ex H being Element of S st e Joins v,w,H
proof
  let S be GraphUnionSet, G be GraphUnion of S;
  let e,v,w be object;
  assume e Joins v,w,G;
  then per cases by GLIB_000:16;
  suppose e DJoins v,w,G;
    then consider H being Element of S such that
      A1: e DJoins v,w,H by Th119;
    take H;
    thus thesis by A1, GLIB_000:16;
  end;
  suppose e DJoins w,v,G;
    then consider H being Element of S such that
      A2: e DJoins w,v,H by Th119;
    take H;
    thus thesis by A2, GLIB_000:16;
  end;
end;
