reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem
  2 <= k implies card primeNumbers(k,10) <= 4
  proof
    set A = primeNumbers(k,10);
    assume 2 <= k;
    then per cases by XXREAL_0:1;
    suppose k = 2;
      hence thesis by Th71,CARD_2:59;
    end;
    suppose
A1:   k > 2;
      consider m being Multiple of 3 such that
A2:   m in seq(k,10) and
A3:   m is odd by Th84;
      set D = multiples(2) /\ seq(k,10);
A4:   now
        assume m in D;
        then reconsider m as Multiple of 2 by Th61;
        2 divides m by Def15;
        hence contradiction by A3;
      end;
      set W = D \/ {m};
      10 = 2*5;
      then card D = 5 by Th83;
      then
A5:   card W = 5 + 1 by A4,CARD_2:41;
A6:   card seq(k,10) = 10 by Th73;
A7:   k+1 > 2+1 by A1,XREAL_1:6;
      now
        assume W meets A;
        then consider x being object such that
A8:     x in W and
A9:     x in A by XBOOLE_0:3;
        reconsider x as Prime by A9,NEWTON:def 6;
A10:    1+k <= x by A9,CALCUL_2:1;
        per cases by A8,XBOOLE_0:def 3;
        suppose x in D;
          then x = 1 or x = 2 by Th62,INT_2:def 4;
          hence contradiction by A7,A10,XXREAL_0:2;
        end;
        suppose x in {m};
          then x = m by TARSKI:def 1;
          hence contradiction by A7,A10,Th60;
        end;
      end;
      then
A11:  card (W \/ A) = 6 + card A by A5,CARD_2:40;
      {m} c= seq(k,10) by A2,ZFMISC_1:31;
      then W c= seq(k,10) by XBOOLE_1:8;
      then W \/ A c= seq(k,10) by XBOOLE_1:8;
      then card A <= 10-6 by A6,A11,XREAL_1:19,NAT_1:43;
      hence thesis;
    end;
  end;
