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 Th47:
  mi(B ^ C) c= mi B ^ C
proof
A1: mi B ^ C c= B ^ C by Th40,Th46;
  now
    let a;
    assume
A2: a in mi(B ^ C);
    then a in B ^ C by Th36;
    then ex b st b c= a & b in mi B ^ C by Lm6;
    hence a in mi B ^ C by A1,A2,Th36;
  end;
  hence thesis by Lm5;
end;
