
theorem
  for S1, S2 being Graph-membered set holds
    the_Vertices_of(S1 /\ S2) c= the_Vertices_of S1 /\ the_Vertices_of S2 &
    the_Edges_of(S1 /\ S2) c= the_Edges_of S1 /\ the_Edges_of S2 &
    the_Source_of(S1 /\ S2) c= the_Source_of S1 /\ the_Source_of S2 &
    the_Target_of(S1 /\ S2) c= the_Target_of S1 /\ the_Target_of S2
proof
  let S1, S2 be Graph-membered set;
  hereby :: vertices
    now
      let x be object;
      assume x in the_Vertices_of(S1 /\ S2);
      then consider G being _Graph such that
        A1: G in S1 /\ S2 & x = the_Vertices_of G by Def14;
      G in S1 & G in S2 by A1, XBOOLE_0:def 4;
      then x in the_Vertices_of S1 & x in the_Vertices_of S2 by A1, Def14;
      hence x in the_Vertices_of S1 /\ the_Vertices_of S2 by XBOOLE_0:def 4;
    end;
    hence the_Vertices_of(S1 /\ S2) c= the_Vertices_of S1 /\ the_Vertices_of S2
      by TARSKI:def 3;
  end;
  hereby :: edges
    now
      let x be object;
      assume x in the_Edges_of(S1 /\ S2);
      then consider G being _Graph such that
        A2: G in S1 /\ S2 & x = the_Edges_of G by Def15;
      G in S1 & G in S2 by A2, XBOOLE_0:def 4;
      then x in the_Edges_of S1 & x in the_Edges_of S2 by A2, Def15;
      hence x in the_Edges_of S1 /\ the_Edges_of S2 by XBOOLE_0:def 4;
    end;
    hence the_Edges_of(S1 /\ S2) c= the_Edges_of S1 /\ the_Edges_of S2
      by TARSKI:def 3;
  end;
  hereby :: sources
    now
      let x be object;
      assume x in the_Source_of(S1 /\ S2);
      then consider G being _Graph such that
        A3: G in S1 /\ S2 & x = the_Source_of G by Def16;
      G in S1 & G in S2 by A3, XBOOLE_0:def 4;
      then x in the_Source_of S1 & x in the_Source_of S2 by A3, Def16;
      hence x in the_Source_of S1 /\ the_Source_of S2 by XBOOLE_0:def 4;
    end;
    hence the_Source_of(S1 /\ S2) c= the_Source_of S1 /\ the_Source_of S2
      by TARSKI:def 3;
  end;
  hereby :: targets
    now
      let x be object;
      assume x in the_Target_of(S1 /\ S2);
      then consider G being _Graph such that
        A4: G in S1 /\ S2 & x = the_Target_of G by Def17;
      G in S1 & G in S2 by A4, XBOOLE_0:def 4;
      then x in the_Target_of S1 & x in the_Target_of S2 by A4, Def17;
      hence x in the_Target_of S1 /\ the_Target_of S2 by XBOOLE_0:def 4;
    end;
    hence the_Target_of(S1 /\ S2) c= the_Target_of S1 /\ the_Target_of S2
      by TARSKI:def 3;
  end;
end;
