reserve X for set,
        D for a_partition of X,
        TG for non empty TopologicalGroup;
reserve A for Subset of X;

theorem
  X = {1,2,3} & A = {1} implies
  [2,1] in [:X \ A,X:] \/ [:X,A:] &
  not [2,1] in [:A,A:] \/ [:X \ A,X \ A:]
  proof
    assume that
A1: X = {1,2,3} and
A2: A = {1};
    1 in X & 2 in X \ A by A1,A2,Th32,TARSKI:def 2,ENUMSET1:def 1;
    then [2,1] in [:X \ A,X:] by ZFMISC_1:87;
    hence [2,1] in [:X \ A,X:] \/ [:X,A:] by XBOOLE_0:def 3;
    assume [2,1] in [:A,A:] \/ [:X\A,X\A:];
    then [2,1] in [:A,A:] or [2,1] in [:X\A,X\A:] by XBOOLE_0:def 3;
    then 2 in {1} or 1 in {2,3} by A1,A2,Th32,ZFMISC_1:87;
    hence thesis by TARSKI:def 1,TARSKI:def 2;
  end;
