 reserve n,i,k,m for Nat;
 reserve p for Prime;

theorem LogAdd:
  for x,y being Real st 0 < x & 0 < y holds
    ln.(x * y) = ln.x + ln.y
  proof
    let x,y be Real;
    assume
A1: 0 < x & 0 < y; then
A2: x in right_open_halfline 0 & y in right_open_halfline 0 by XXREAL_1:235;
a2: x * y in right_open_halfline 0 by XXREAL_1:235,A1;
A3: number_e > 1 by XXREAL_0:2,TAYLOR_1:11;
    ln.(x * y) = log (number_e,x * y) by a2,TAYLOR_1:def 2,def 3
       .= log (number_e,x) + log(number_e,y) by POWER:53,A1,A3
       .= log (number_e,x) + (log_number_e).y by TAYLOR_1:def 2,A2
       .= ln.x + ln.y by TAYLOR_1:def 3,def 2,A2;
    hence thesis;
  end;
