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 < b & c < 0 implies a to_power c > b to_power c
proof
  assume that
A1: a > 0 and
A2: a < b and
A3: c < 0;
A4: a to_power c > 0 by A1,Th34;
A5: a to_power c <> 0 by A1,Th34;
 a/a<b/a by A1,A2,XREAL_1:74;
then  1<b/a by A1,XCMPLX_1:60;
then  (b/a) to_power c < 1 by A3,Th36;
then  b to_power c / a to_power c < 1 by A1,A2,Th31;
then
 b to_power c/a to_power c*a to_power c <1*a to_power c by A4,XREAL_1:68;
  hence thesis by A5,XCMPLX_1:87;
end;
