
theorem Th7:
  for S being Graph-membered set holds
    card the_Vertices_of S c= card S &
    card the_Edges_of S c= card S &
    card the_Source_of S c= card S &
    card the_Target_of S c= card S
proof
  let S be Graph-membered set;
  :: first the vertices
  defpred P1[object,object] means ex G being _Graph st
    $1 = G & $2 = the_Vertices_of G;
  A1: for x,y1,y2 being object st x in S & P1[x,y1] & P1[x,y2] holds y1 = y2;
  A2: for x being object st x in S ex y being object st P1[x,y]
  proof
    let x be object;
    assume x in S;
    then reconsider G = x as _Graph;
    take the_Vertices_of G, G;
    thus thesis;
  end;
  consider f1 being Function such that
    A3: dom f1 = S & for x being object st x in S holds P1[x,f1.x]
    from FUNCT_1:sch 2(A1,A2);
  now
    let x being object;
    assume x in the_Vertices_of S;
    then consider G being _Graph such that
      A4: G in S & x = the_Vertices_of G by Def14;
    consider G0 being _Graph such that
      A5: G = G0 & f1.G = the_Vertices_of G0 by A3, A4;
    thus x in rng f1 by A3, A4, A5, FUNCT_1:3;
  end;
  hence card the_Vertices_of S c= card S by A3, CARD_1:12, TARSKI:def 3;
  :: the other three cases work analogously
  defpred P2[object,object] means ex G being _Graph st
    $1 = G & $2 = the_Edges_of G;
  A6: for x,y1,y2 being object st x in S & P2[x,y1] & P2[x,y2] holds y1 = y2;
  A7: for x being object st x in S ex y being object st P2[x,y]
  proof
    let x be object;
    assume x in S;
    then reconsider G = x as _Graph;
    take the_Edges_of G, G;
    thus thesis;
  end;
  consider f2 being Function such that
    A8: dom f2 = S & for x being object st x in S holds P2[x,f2.x]
    from FUNCT_1:sch 2(A6,A7);
  now
    let x being object;
    assume x in the_Edges_of S;
    then consider G being _Graph such that
      A9: G in S & x = the_Edges_of G by Def15;
    consider G0 being _Graph such that
      A10: G = G0 & f2.G = the_Edges_of G0 by A8, A9;
    thus x in rng f2 by A8, A9, A10, FUNCT_1:3;
  end;
  hence card the_Edges_of S c= card S by A8, CARD_1:12, TARSKI:def 3;
  defpred P3[object,object] means ex G being _Graph st
    $1 = G & $2 = the_Source_of G;
  A11: for x,y1,y2 being object st x in S & P3[x,y1] & P3[x,y2] holds y1 = y2;
  A12: for x being object st x in S ex y being object st P3[x,y]
  proof
    let x be object;
    assume x in S;
    then reconsider G = x as _Graph;
    take the_Source_of G, G;
    thus thesis;
  end;
  consider f3 being Function such that
    A13: dom f3 = S & for x being object st x in S holds P3[x,f3.x]
    from FUNCT_1:sch 2(A11,A12);
  now
    let x being object;
    assume x in the_Source_of S;
    then consider G being _Graph such that
      A14: G in S & x = the_Source_of G by Def16;
    consider G0 being _Graph such that
      A15: G = G0 & f3.G = the_Source_of G0 by A13, A14;
    thus x in rng f3 by A13, A14, A15, FUNCT_1:3;
  end;
  hence card the_Source_of S c= card S by A13, CARD_1:12, TARSKI:def 3;
  defpred P4[object,object] means ex G being _Graph st
    $1 = G & $2 = the_Target_of G;
  A16: for x,y1,y2 being object st x in S & P4[x,y1] & P4[x,y2] holds y1 = y2;
  A17: for x being object st x in S ex y being object st P4[x,y]
  proof
    let x be object;
    assume x in S;
    then reconsider G = x as _Graph;
    take the_Target_of G, G;
    thus thesis;
  end;
  consider f4 being Function such that
    A18: dom f4 = S & for x being object st x in S holds P4[x,f4.x]
    from FUNCT_1:sch 2(A16,A17);
  now
    let x being object;
    assume x in the_Target_of S;
    then consider G being _Graph such that
      A19: G in S & x = the_Target_of G by Def17;
    consider G0 being _Graph such that
      A20: G = G0 & f4.G = the_Target_of G0 by A18, A19;
    thus x in rng f4 by A18, A19, A20, FUNCT_1:3;
  end;
  hence card the_Target_of S c= card S by A18, CARD_1:12, TARSKI:def 3;
end;
