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

theorem Th3:
  A1,A2 constitute_a_decomposition implies A1 = A2` & A2 = A1`
proof
  assume
A1: A1,A2 constitute_a_decomposition;
  then
A2: A1 misses A2;
  then
A3: A1 c= A2` by SUBSET_1:23;
A4: A2 c= A1` by A2,SUBSET_1:23;
A5: A1 \/ A2 = [#]X by A1;
  then A2` c= A1 by TDLAT_1:1;
  hence A1 = A2` by A3;
  A1` c= A2 by A5,TDLAT_1:1;
  hence thesis by A4;
end;
