 reserve n,m,i,p for Nat,
         h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
 reserve f,f1,f2,g for Function of REAL,REAL;

theorem Th4:
  x0>0 & x1>0 implies ln.x0 - ln.x1 = ln.(x0/x1)
proof
  assume
A1: x0>0 & x1>0;
A2: log(number_e,x0/x1) = ln.(x0/x1)
  proof
    x0/x1 in right_open_halfline(0)
  proof
    x0/x1 in {g where g is Real: 0<g} by A1;
    hence thesis by XXREAL_1:230;
  end;
  hence thesis by TAYLOR_1:def 2;
  end;
  ln.x0-ln.x1 = log(number_e,x0)-ln.x1 by A1,Th3
    .= log(number_e,x0)-log(number_e,x1) by A1,Th3
    .= log(number_e,x0/x1) by A1,Th1;
  hence thesis by A2;
end;
