
theorem
  for a,b be positive Real, n be non zero Real holds
  a to_power n = b to_power n iff a = b
  proof
    let a,b be positive Real, n be non zero Real;
    a <> b implies a to_power n <> b to_power n
    proof
      assume a <> b; then
      per cases by XXREAL_0:1;
      suppose
        B1: a > b;
        per cases;
        suppose n > 0;
          hence thesis by B1,POWER:37;
        end;
        suppose n < 0;
          hence thesis by B1,POWER:38;
        end;
      end;
      suppose
        B1: a < b;
        per cases;
        suppose n > 0;
          hence thesis by B1,POWER:37;
        end;
        suppose n < 0;
          hence thesis by B1,POWER:38;
        end;
      end;
    end;
    hence thesis;
  end;
