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
  n <= 341 & n divides 2|^n-2 & not n divides 3|^n-3 implies n = 341
  proof
    assume n <= 341;
    then per cases by XXREAL_0:1;
    suppose n = 341;
      hence thesis;
    end;
    suppose
A1:   n < 341;
      assume that
A2:   n divides 2|^n-2 and
A3:   not n divides 3|^n-3;
      n is composite by A2,A3,Th62;
      then n in {341,561,645,1105} by A1,A2,Th49,XXREAL_0:2;
      then n = 341 or n = 561 or n = 645 or n = 1105 by ENUMSET1:def 2;
      hence thesis by A1;
    end;
  end;
