reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;

theorem Th34:
  a|^n is prime implies n = 1
  proof
    set p = a|^n;
    assume that
A1: p > 1 and
A2: for n being Nat st n divides p holds n = 1 or n = p;
    per cases by NAT_1:6;
    suppose n = 0;
      hence thesis by A1,NEWTON:4;
    end;
    suppose ex k st n = k+1;
      then consider k such that
A3:   n = k+1;
A4:   p = a|^k*a by A3,NEWTON:6;
A5:   k in NAT by ORDINAL1:def 12;
      a divides a|^k*a;
      then per cases by A2,A4;
      suppose a = 1;
        hence thesis by A1;
      end;
      suppose a = p;
        then 1*a = a|^k*a by A3,NEWTON:6;
        then k = 0 by A1,A4,A5,WEDDWITT:1,XCMPLX_1:5;
        hence thesis by A3;
      end;
    end;
  end;
