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

theorem
  for p being Prime holds
  p <= k implies multiples(p) /\ seq(k,m) misses primeNumbers(k,m)
  proof
    let p be Prime;
    set A = primeNumbers(k,m);
    set D = multiples(p) /\ seq(k,m);
    assume
A1: p <= k;
    assume D meets A;
    then consider x being object such that
A2: x in D and
A3: x in A by XBOOLE_0:3;
A4: x is Multiple of p by A2,Th61;
    x in seq(k,m) by A2;
    then consider n being Element of NAT such that
A5: x = n and
A6: k+1 <= n and n <= k+m;
A7: p+1 <= k+1 by A1,XREAL_1:6;
    n is prime by A3,A5,NEWTON:def 6;
    then p = 1 or p = n by A4,A5,Def15;
    then p+1 <= p+0 by A6,A7,XXREAL_0:2,INT_2:def 4;
    hence thesis by XREAL_1:6;
  end;
