reserve x for set;
reserve a, b, c, d, e for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p for Rational;

theorem
  a>0 & a<>1 & b>0 & c>0 implies log(a,b) - log(a,c) = log(a,b/c)
proof
  assume that
A1: a>0 and
A2: a<>1 and
A3: b>0 and
A4: c>0;
A5: a to_power (log(a,b) - log(a,c))
  = a to_power log(a,b) / a to_power log(a,c) by A1,Th29
    .= b / a to_power log(a,c) by A1,A2,A3,Def3
    .= b / c by A1,A2,A4,Def3;
 b/c>0 by A3,A4,XREAL_1:139;
  hence thesis by A1,A2,A5,Def3;
end;
