theorem Th17:
  mi(A ^ B) c= mi A ^ B
proof
A1: mi A ^ B c= A ^ B by Th8,Th14;
  now
    let a be set;
    assume
A2: a in mi (A ^ B);
    then a in A ^ B & a is finite by Lm1,Th6;
    then ex b be finite set st b c= a & b in mi A ^ B by Lm2;
    hence a in mi A ^ B by A1,A2,Th6;
  end;
  hence thesis;
end;
