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;
