reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X for TopSpace;
reserve A1, A2 for Subset of X;
reserve A1,A2 for Subset of X;

theorem Th40:
  for B being Subset of X holds A1,B are_separated or A2,B
  are_separated implies A1 /\ A2,B are_separated
proof
  let B be Subset of X;
  thus A1,B are_separated or A2,B are_separated implies A1 /\ A2,B
  are_separated
  proof
A1: now
A2:   A1 /\ A2 c= A2 by XBOOLE_1:17;
      assume A2,B are_separated;
      hence thesis by A2,CONNSP_1:7;
    end;
A3: now
A4:   A1 /\ A2 c= A1 by XBOOLE_1:17;
      assume A1,B are_separated;
      hence thesis by A4,CONNSP_1:7;
    end;
    assume A1,B are_separated or A2,B are_separated;
    hence thesis by A3,A1;
  end;
end;
