reserve V, C for set;
reserve A, B, D for Element of Fin PFuncs (V, C);
reserve s for Element of PFuncs (V,C);

theorem Th19:
  mi(mi A ^ B) = mi (A ^ B)
proof
A1: mi A ^ B c= A ^ B by Th8,Th14;
  now
    let a be set;
    assume
A2: a in mi (mi A ^ B);
A3: now
      let b be finite set;
      assume b in A ^ B;
      then consider c be finite set such that
A4:   c c= b and
A5:   c in mi A ^ B by Lm2;
      assume
A6:   b c= a;
      then c c= a by A4;
      then c = a by A2,A5,Th6;
      hence b = a by A4,A6;
    end;
    a in mi A ^ B & a is finite by A2,Lm1,Th6;
    hence a in mi (A ^ B) by A1,A3,Th7;
  end;
  hence mi(mi A ^ B) c= mi(A ^ B);
A7: mi(A ^ B) c= mi A ^ B by Th17;
  now
    let a be set;
    assume
A8: a in mi(A ^ B);
    then a is finite & for b be finite set st b in mi A ^ B holds b c= a
    implies b = a by A1,Lm1,Th6;
    hence a in mi(mi A ^ B) by A7,A8,Th7;
  end;
  hence thesis;
end;
