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 Th42:
  mi K = K
proof
  thus mi K c= K by Th40;
  now
    let a;
    assume
A1: a in K;
    then for b st b in K & b c= a holds b = a by Th32;
    hence a in mi K by A1,Th39;
  end;
  hence thesis by Lm5;
end;
