
theorem Th6:
  for G, H being _Graph holds
    the_Vertices_of {G,H} = {the_Vertices_of G, the_Vertices_of H} &
    the_Edges_of {G,H} = {the_Edges_of G, the_Edges_of H} &
    the_Source_of {G,H} = {the_Source_of G, the_Source_of H} &
    the_Target_of {G,H} = {the_Target_of G, the_Target_of H}
proof
  let G, H be _Graph;
  now
    let x be object;
    hereby
      assume x in the_Vertices_of {G,H};
      then consider G0 being Element of {G,H} such that
        A1: x = the_Vertices_of G0;
      per cases by TARSKI:def 2;
      suppose G0 = G;
        hence x in {the_Vertices_of G, the_Vertices_of H} by A1, TARSKI:def 2;
      end;
      suppose G0 = H;
        hence x in {the_Vertices_of G, the_Vertices_of H} by A1, TARSKI:def 2;
      end;
    end;
    assume x in {the_Vertices_of G, the_Vertices_of H};
    then A2: x = the_Vertices_of G or x = the_Vertices_of H by TARSKI:def 2;
    G in {G,H} & H in {G,H} by TARSKI:def 2;
    hence x in the_Vertices_of {G,H} by A2;
  end;
  hence the_Vertices_of {G,H} = {the_Vertices_of G, the_Vertices_of H}
    by TARSKI:2;
  now
    let x be object;
    hereby
      assume x in the_Edges_of {G,H};
      then consider G0 being Element of {G,H} such that
        A3: x = the_Edges_of G0;
      per cases by TARSKI:def 2;
      suppose G0 = G;
        hence x in {the_Edges_of G, the_Edges_of H} by A3, TARSKI:def 2;
      end;
      suppose G0 = H;
        hence x in {the_Edges_of G, the_Edges_of H} by A3, TARSKI:def 2;
      end;
    end;
    assume x in {the_Edges_of G, the_Edges_of H};
    then A4: x = the_Edges_of G or x = the_Edges_of H by TARSKI:def 2;
    G in {G,H} & H in {G,H} by TARSKI:def 2;
    hence x in the_Edges_of {G,H} by A4;
  end;
  hence the_Edges_of {G,H} = {the_Edges_of G, the_Edges_of H}
    by TARSKI:2;
  now
    let x be object;
    hereby
      assume x in the_Source_of {G,H};
      then consider G0 being Element of {G,H} such that
        A5: x = the_Source_of G0;
      per cases by TARSKI:def 2;
      suppose G0 = G;
        hence x in {the_Source_of G, the_Source_of H} by A5, TARSKI:def 2;
      end;
      suppose G0 = H;
        hence x in {the_Source_of G, the_Source_of H} by A5, TARSKI:def 2;
      end;
    end;
    assume x in {the_Source_of G, the_Source_of H};
    then A6: x = the_Source_of G or x = the_Source_of H by TARSKI:def 2;
    G in {G,H} & H in {G,H} by TARSKI:def 2;
    hence x in the_Source_of {G,H} by A6;
  end;
  hence the_Source_of {G,H} = {the_Source_of G, the_Source_of H}
    by TARSKI:2;
  now
    let x be object;
    hereby
      assume x in the_Target_of {G,H};
      then consider G0 being Element of {G,H} such that
        A7: x = the_Target_of G0;
      per cases by TARSKI:def 2;
      suppose G0 = G;
        hence x in {the_Target_of G, the_Target_of H} by A7, TARSKI:def 2;
      end;
      suppose G0 = H;
        hence x in {the_Target_of G, the_Target_of H} by A7, TARSKI:def 2;
      end;
    end;
    assume x in {the_Target_of G, the_Target_of H};
    then A8: x = the_Target_of G or x = the_Target_of H by TARSKI:def 2;
    G in {G,H} & H in {G,H} by TARSKI:def 2;
    hence x in the_Target_of {G,H} by A8;
  end;
  hence the_Target_of {G,H} = {the_Target_of G, the_Target_of H}
    by TARSKI:2;
end;
