
theorem
  for a,m,n be positive Real holds
  a to_power n + a to_power m =
    (a to_power min(n,m))*(1 + (a to_power |.m-n.|))
  proof
    let a,m,n be positive Real;
    per cases;
    suppose
      B1: n >= m; then
      n - m >= m - m by XREAL_1:9; then
      reconsider k = n - m as non negative Real;
      a to_power n = a to_power (m+k)
      .= (a to_power m)*(a to_power k) by POWER:27; then
      a to_power n + a to_power m = (a to_power m)*(1 + a to_power |.-(m-n).|)
      .= (a to_power m)*(1 + a to_power |.m - n.|) by COMPLEX1:52;
      hence thesis by B1,XXREAL_0:def 9;
    end;
    suppose
      B1: n < m; then
      n - n < m - n by XREAL_1:9; then
      reconsider k = m - n as positive Real;
      a to_power m = a to_power (n + k)
      .= (a to_power n)*(a to_power k) by POWER:27; then
      a to_power n + a to_power m = (a to_power n)*(1 + a to_power |.m-n.|);
      hence thesis by B1,XXREAL_0:def 9;
    end;
  end;
