reserve x, y, z, v for set,
  n, m, k for Nat;
reserve G, G1, G2, G3 for Graph;
reserve x, y for Element of (the carrier of G);

theorem
  for G1 being strict Graph holds G1 c= G2 iff bool G1 c= bool G2
proof
  let G1 be strict Graph;
  thus G1 c= G2 implies bool G1 c= bool G2
  proof
    assume
A1: G1 c= G2;
      let x be object;
      assume x in bool G1;
      then reconsider G = x as strict Subgraph of G1 by Def25;
   G c= G1;
then    G c= G2 by A1,Th17;
then    G is strict Subgraph of G2;
      hence thesis by Def25;
  end;
  assume
A2: bool G1 c= bool G2;
 G1 in bool G1 by Th29;
then  G1 is Subgraph of G2 by A2,Def25;
  hence thesis;
end;
