
theorem Th10:
  for S1, S2 being Graph-membered set holds
    the_Vertices_of S1 \ the_Vertices_of S2 c= the_Vertices_of(S1 \ S2) &
    the_Edges_of S1 \ the_Edges_of S2 c= the_Edges_of(S1 \ S2) &
    the_Source_of S1 \ the_Source_of S2 c= the_Source_of(S1 \ S2) &
    the_Target_of S1 \ the_Target_of S2 c= the_Target_of(S1 \ S2)
proof
  let S1, S2 be Graph-membered set;
  hereby :: vertices
    now
      let x be object;
      assume x in the_Vertices_of S1 \ the_Vertices_of S2;
      then A1: x in the_Vertices_of S1 & not x in the_Vertices_of S2
        by XBOOLE_0:def 5;
      then consider G being _Graph such that
        A2: G in S1 & x = the_Vertices_of G by Def14;
       not G in S2 by A1, A2, Def14;
       then G in S1 \ S2 by A2, XBOOLE_0:def 5;
       hence x in the_Vertices_of(S1\S2) by A2, Def14;
    end;
    hence the_Vertices_of S1 \ the_Vertices_of S2 c= the_Vertices_of(S1 \ S2)
      by TARSKI:def 3;
  end;
  hereby :: edges
    now
      let x be object;
      assume x in the_Edges_of S1 \ the_Edges_of S2;
      then A3: x in the_Edges_of S1 & not x in the_Edges_of S2
        by XBOOLE_0:def 5;
      then consider G being _Graph such that
        A4: G in S1 & x = the_Edges_of G by Def15;
       not G in S2 by A3, A4, Def15;
       then G in S1 \ S2 by A4, XBOOLE_0:def 5;
       hence x in the_Edges_of(S1\S2) by A4, Def15;
    end;
    hence the_Edges_of S1 \ the_Edges_of S2 c= the_Edges_of(S1 \ S2)
      by TARSKI:def 3;
  end;
  hereby :: sources
    now
      let x be object;
      assume x in the_Source_of S1 \ the_Source_of S2;
      then A5: x in the_Source_of S1 & not x in the_Source_of S2
        by XBOOLE_0:def 5;
      then consider G being _Graph such that
        A6: G in S1 & x = the_Source_of G by Def16;
       not G in S2 by A5, A6, Def16;
       then G in S1 \ S2 by A6, XBOOLE_0:def 5;
       hence x in the_Source_of(S1\S2) by A6, Def16;
    end;
    hence the_Source_of S1 \ the_Source_of S2 c= the_Source_of(S1 \ S2)
      by TARSKI:def 3;
  end;
  hereby :: targets
    now
      let x be object;
      assume x in the_Target_of S1 \ the_Target_of S2;
      then A7: x in the_Target_of S1 & not x in the_Target_of S2
        by XBOOLE_0:def 5;
      then consider G being _Graph such that
        A8: G in S1 & x = the_Target_of G by Def17;
       not G in S2 by A7, A8, Def17;
       then G in S1 \ S2 by A8, XBOOLE_0:def 5;
       hence x in the_Target_of(S1\S2) by A8, Def17;
    end;
    hence the_Target_of S1 \ the_Target_of S2 c= the_Target_of(S1 \ S2)
      by TARSKI:def 3;
  end;
end;
