reserve X for non empty 1-sorted;
reserve A, A1, A2, B1, B2 for Subset of X;

theorem Th7:
  A,A do_not_constitute_a_decomposition
proof
  assume
A1: A,A constitute_a_decomposition;
  per cases;
  suppose
A2: A is non empty;
    A = A` by A1,Th3;
    then A misses A by SUBSET_1:23;
    then A /\ A = {};
    hence contradiction by A2;
  end;
  suppose
    A is empty;
    then {} = A \/ A .= the carrier of X by A1;
    hence contradiction;
  end;
end;
