reserve d,i,j,k,m,n,p,q,x,k1,k2 for Nat,
  a,c,i1,i2,i3,i5 for Integer;

theorem Th30:
  k > 1 & k |^ n = k |^ m implies n = m
proof
  assume that
A1: k > 1 and
A2: k |^ n = k |^ m;
  now
    per cases;
    suppose
      n = m;
      hence thesis;
    end;
    suppose
      n <> m;
      then k to_power m <> k to_power n by A1,POWER:50;
      hence thesis by A2;
    end;
  end;
  hence thesis;
end;
