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 Th35:
  1 < a implies ex k st 1 < k & n < a|^k
  proof
    assume
A1: 1 < a;
    per cases by NAT_1:14;
    suppose
A2:   n = 0;
      take 2;
      thus thesis by A1,A2;
    end;
    suppose n >= 1;
      then per cases by XXREAL_0:1;
      suppose
A3:     n = 1;
        take 2;
        thus 1 < 2;
        1*1 < a*a by A1,XREAL_1:96;
        hence thesis by A3,WSIERP_1:1;
      end;
      suppose
A4:     n > 1;
        take n;
        thus thesis by A1,A4,NAT_3:2;
      end;
    end;
  end;
