reserve n, k, r, m, i, j for Nat;

theorem Th49:
  for n being Element of NAT st n > 1 & n <> 4 holds n is non
prime implies ex k being non zero Element of NAT st k <> 1 & k <> 2 & k <> n &
  k divides n
proof
  let n be Element of NAT such that
A1: n > 1 and
A2: n <> 4;
  assume
A3: n is non prime;
  per cases by A3,INT_2:def 4;
  suppose
    n <= 1;
    hence thesis by A1;
  end;
  suppose
    not for k being Nat holds k divides n implies k = 1 or k = n;
    then consider k being Nat such that
A4: k divides n and
A5: k <> 1 & k <> n;
    consider m being Nat such that
A6: n = k*m by A4,NAT_D:def 3;
A7: m divides n & m is non zero Element of NAT by A1,A6,NAT_D:def 3
,ORDINAL1:def 12;
A8: k is non zero Element of NAT by A1,A4,INT_2:3,ORDINAL1:def 12;
A9: k <> 2 or m <> 2 by A2,A6;
    m <> 1 & m <> n by A1,A5,A6,XCMPLX_1:7;
    hence thesis by A4,A5,A8,A7,A9;
  end;
end;
