
theorem
  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;
  the_Vertices_of S1 \ the_Vertices_of S2 c= the_Vertices_of(S1\S2) &
    the_Vertices_of S2 \ the_Vertices_of S1 c= the_Vertices_of(S2\S1) by Th10;
  then the_Vertices_of S1 \+\ the_Vertices_of S2
    c= the_Vertices_of(S1\S2) \/ the_Vertices_of(S2\S1) by XBOOLE_1:13;
  hence the_Vertices_of S1 \+\ the_Vertices_of S2 c= the_Vertices_of(S1 \+\ S2)
    by Th8;
  the_Edges_of S1 \ the_Edges_of S2 c= the_Edges_of(S1\S2) &
    the_Edges_of S2 \ the_Edges_of S1 c= the_Edges_of(S2\S1) by Th10;
  then the_Edges_of S1 \+\ the_Edges_of S2
    c= the_Edges_of(S1\S2) \/ the_Edges_of(S2\S1) by XBOOLE_1:13;
  hence the_Edges_of S1 \+\ the_Edges_of S2 c= the_Edges_of(S1 \+\ S2) by Th8;
  the_Source_of S1 \ the_Source_of S2 c= the_Source_of(S1\S2) &
    the_Source_of S2 \ the_Source_of S1 c= the_Source_of(S2\S1) by Th10;
  then the_Source_of S1 \+\ the_Source_of S2
    c= the_Source_of(S1\S2) \/ the_Source_of(S2\S1) by XBOOLE_1:13;
  hence the_Source_of S1 \+\ the_Source_of S2 c= the_Source_of(S1\+\S2) by Th8;
  the_Target_of S1 \ the_Target_of S2 c= the_Target_of(S1\S2) &
    the_Target_of S2 \ the_Target_of S1 c= the_Target_of(S2\S1) by Th10;
  then the_Target_of S1 \+\ the_Target_of S2
    c= the_Target_of(S1\S2) \/ the_Target_of(S2\S1) by XBOOLE_1:13;
  hence the_Target_of S1 \+\ the_Target_of S2 c= the_Target_of(S1\+\S2) by Th8;
end;
