reserve G, G1, G2 for _Graph, H for Subgraph of G;

theorem Th199:
  (G is loopless iff G.allComponents() is loopless) &
  (G is non-multi iff G.allComponents() is non-multi) &
  (G is non-Dmulti iff G.allComponents() is non-Dmulti) &
  (G is simple iff G.allComponents() is simple) &
  (G is Dsimple iff G.allComponents() is Dsimple) &
  (G is acyclic iff G.allComponents() is acyclic) &
  (G is edgeless iff G.allComponents() is edgeless) &
  (G is chordal iff G.allComponents() is chordal) &
  (G is loopfull iff G.allComponents() is loopfull)
proof
  A1: G is GraphUnion of G.allComponents() by Th198;
  thus G is loopless iff G.allComponents() is loopless by A1;
  thus G is non-multi iff G.allComponents() is non-multi by A1;
  thus G is non-Dmulti iff G.allComponents() is non-Dmulti by A1;
  thus G is simple iff G.allComponents() is simple by A1;
  thus G is Dsimple iff G.allComponents() is Dsimple by A1;
  thus G is acyclic iff G.allComponents() is acyclic by A1;
  thus G is edgeless iff G.allComponents() is edgeless by A1;
  thus G is chordal iff G.allComponents() is chordal by A1 ,GLIB_015:63;
  thus G is loopfull iff G.allComponents() is loopfull by A1, GLIB_015:63;
end;
