
theorem
  for S being vertex-disjoint GraphUnionSet, G being GraphUnion of S holds
    (S is simple iff G is simple) &
    (S is Dsimple iff G is Dsimple)
proof
  let S be vertex-disjoint GraphUnionSet, G be GraphUnion of S;
  hereby
    assume S is simple;
    then G is loopless non-multi by Th60;
    hence G is simple;
  end;
  hereby
    assume G is simple;
    then G is loopless non-multi;
    then S is loopless non-multi by Th60, GLIB_014:23;
    hence S is simple;
  end;
  hereby
    assume S is Dsimple;
    then G is loopless non-Dmulti by Th60;
    hence G is Dsimple;
  end;
  hereby
    assume G is Dsimple;
    then S is loopless non-Dmulti by Th60, GLIB_014:23;
    hence S is Dsimple;
  end;
end;
