reserve A, B for non empty preBoolean set,
  x, y for Element of [:A,B:];
reserve X for set,
  a,b,c for Element of [:A,B:];
reserve a for Element of [:Fin X, Fin X:];
reserve A for set;
reserve x,y for Element of [:Fin X, Fin X:],
  a,b for Element of DISJOINT_PAIRS X;
reserve A for set,
  x for Element of [:Fin A, Fin A:],
  a,b,c,d,s,t for Element of DISJOINT_PAIRS A,
  B,C,D for Element of Fin DISJOINT_PAIRS A;
reserve K,L,M for Element of Normal_forms_on A;

theorem Th53:
  K^(L \/ M) = K^L \/ K^M
proof
  now
    let a;
    assume a in K^(L \/ M);
    then consider b,c such that
A1: b in K and
A2: c in L \/ M and
A3: a = b \/ c by Th34;
    c in L or c in M by A2,XBOOLE_0:def 3;
    then a in K^L or a in K^M by A1,A3,Th35;
    hence a in K^L \/ K^M by XBOOLE_0:def 3;
  end;
  hence K^(L \/ M) c= K^L \/ K^M by Lm5;
  K^L c= K^(L \/ M) & K^M c= K^(L \/ M) by Th49,XBOOLE_1:7;
  hence thesis by XBOOLE_1:8;
end;
