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 Th12:
  mi (A \/ B) c= mi A \/ B
proof
  now
    let a be set;
    assume
A1: a in mi (A \/ B);
    then reconsider a9 = a as finite set by Lm1;
A2: a in A \/ B by A1,Th6;
    now
      per cases by A2,XBOOLE_0:def 3;
      suppose
A3:     a in A;
        now
          let b be finite set;
          assume b in A;
          then b in A \/ B by XBOOLE_0:def 3;
          hence b c= a implies b = a by A1,Th6;
        end;
        then a9 in mi A by A3,Th7;
        hence a in mi A \/ B by XBOOLE_0:def 3;
      end;
      suppose
        a in B;
        hence a in mi A \/ B by XBOOLE_0:def 3;
      end;
    end;
    hence a in mi A \/ B;
  end;
  hence thesis;
end;
