
theorem Th5:
  for G being _Graph holds
    the_Vertices_of {G} = {the_Vertices_of G} &
    the_Edges_of {G} = {the_Edges_of G} &
    the_Source_of {G} = {the_Source_of G} &
    the_Target_of {G} = {the_Target_of G}
proof
  let G be _Graph;
  now
    let x be object;
    hereby
      assume x in the_Vertices_of {G};
      then consider G0 being Element of {G} such that
        A1: x = the_Vertices_of G0;
      thus x = the_Vertices_of G by A1, TARSKI:def 1;
    end;
    assume A2: x = the_Vertices_of G;
    G in {G} by TARSKI:def 1;
    hence x in the_Vertices_of {G} by A2;
  end;
  hence the_Vertices_of {G} = {the_Vertices_of G} by TARSKI:def 1;
  now
    let x be object;
    hereby
      assume x in the_Edges_of {G};
      then consider G0 being Element of {G} such that
        A3: x = the_Edges_of G0;
      thus x = the_Edges_of G by A3, TARSKI:def 1;
    end;
    assume A4: x = the_Edges_of G;
    G in {G} by TARSKI:def 1;
    hence x in the_Edges_of {G} by A4;
  end;
  hence the_Edges_of {G} = {the_Edges_of G} by TARSKI:def 1;
  now
    let x be object;
    hereby
      assume x in the_Source_of {G};
      then consider G0 being Element of {G} such that
        A5: x = the_Source_of G0;
      thus x = the_Source_of G by A5, TARSKI:def 1;
    end;
    assume A6: x = the_Source_of G;
    G in {G} by TARSKI:def 1;
    hence x in the_Source_of {G} by A6;
  end;
  hence the_Source_of {G} = {the_Source_of G} by TARSKI:def 1;
  now
    let x be object;
    hereby
      assume x in the_Target_of {G};
      then consider G0 being Element of {G} such that
        A7: x = the_Target_of G0;
      thus x = the_Target_of G by A7, TARSKI:def 1;
    end;
    assume A8: x = the_Target_of G;
    G in {G} by TARSKI:def 1;
    hence x in the_Target_of {G} by A8;
  end;
  hence the_Target_of {G} = {the_Target_of G} by TARSKI:def 1;
end;
