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 Th13:
  mi(mi A \/ B) = mi (A \/ B)
proof
A1: mi A \/ B c= A \/ B by Th8,XBOOLE_1:9;
  now
    let a be set;
    assume
A2: a in mi(mi A \/ B);
    then reconsider a9 = a as finite set by Lm1;
A3: now
      let b be finite set;
      assume that
A4:   b in A \/ B and
A5:   b c= a;
      now
        per cases by A4,XBOOLE_0:def 3;
        suppose
          b in A;
          then consider c be set such that
A6:       c c= b and
A7:       c in mi A by Th10;
          c in mi A \/ B & c c= a by A5,A6,A7,XBOOLE_0:def 3;
          then c = a by A2,Th6;
          hence b = a by A5,A6;
        end;
        suppose
          b in B;
          then b in mi A \/ B by XBOOLE_0:def 3;
          hence b = a by A2,A5,Th6;
        end;
      end;
      hence b = a;
    end;
    a in mi A \/ B by A2,Th6;
    then a9 in mi (A \/ B) by A1,A3,Th7;
    hence a in mi (A \/ B);
  end;
  hence mi(mi A \/ B) c= mi (A \/ B);
A8: mi(A \/ B) c= mi A \/ B by Th12;
  now
    let a be set;
    assume
A9: a in mi (A \/ B);
    then reconsider a9 = a as finite set by Lm1;
    for b be finite set st b in mi A \/ B holds b c= a implies b = a by A1,A9
,Th6;
    then a9 in mi(mi A \/ B) by A8,A9,Th7;
    hence a in mi(mi A \/ B);
  end;
  hence thesis;
end;
