reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i for Integer;
reserve r for Real;
reserve p for Prime;

theorem Th62:
  n divides 2|^n-2 & not n divides 3|^n-3 implies n is composite
  proof
    assume that
A1: n divides 2|^n-2 and
A2: not n divides 3|^n-3;
A3: not 3|^n,3 are_congruent_mod n by A2;
    assume not n is composite;
    then per cases by NUMBER02:def 1;
    suppose 1+1 > n;
      then n <= 0+1 by NAT_1:13;
      then n = 0 or ... or n = 1;
      hence contradiction by A1,A2,NEWTON:4;
    end;
    suppose that
A4:   2 <= n and
A5:   n is prime;
      3|^n mod n <> 3 mod n by A3,A4,NAT_D:64;
      hence contradiction by A5,EULER_2:19;
    end;
  end;
