reserve i1 for Element of INT;
reserve j1,j2,j3 for Integer;
reserve p,s,k,n for Nat;
reserve x,y,xp,yp for set;
reserve G for Group;
reserve a,b for Element of G;
reserve F for FinSequence of G;
reserve I for FinSequence of INT;

theorem
  for G being strict finite Group st card G is prime holds G is cyclic
proof
  let G be strict finite Group;
  assume A1: card G is prime;
  set p = card G;
  ex a being Element of G st ord a = p
  proof
    assume
A2: for a being Element of G holds ord a <> p;
A3: now
      let a be Element of G;
      ord a divides p by Th8;
      then ord a = 1 or ord a = p by A1,INT_2:def 4;
      hence ord a = 1 by A2;
    end;
    for x being object
holds x in the carrier of G implies x in the carrier of (1).G
    proof
      let x be object;
      assume x in the carrier of G;
      then reconsider x9=x as Element of G;
      ord x9 = 1 by A3;
      then x9 = 1_G by GROUP_1:43;
      then x9 in {1_G} by TARSKI:def 1;
      hence thesis by GROUP_2:def 7;
    end;
    then the carrier of G c= the carrier of (1).G;
    then G = (1).G by GROUP_2:61;
    then card G = 1 by GROUP_2:69;
    hence contradiction by A1,INT_2:def 4;
  end;
  hence thesis by Th19;
end;
