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 Th40:
  a < b & c > 0 & c < 1 implies c to_power a > c to_power b
proof
  assume that
A1: a < b and
A2: c > 0 and
A3: c < 1;
A4: (1/c) to_power a > 0 by A2,Th34;
A5: (1/c) to_power a <> 0 by A2,Th34;
A6: c to_power a > 0 by A2,Th34;
 c/c <1/c by A2,A3,XREAL_1:74;
then A7: 1<1/c by A2,XCMPLX_1:60;
 b-a>0 by A1,XREAL_1:50;
then  (1/c) to_power (b-a) > 1 by A7,Th35;
then  (1/c) to_power b / (1/c) to_power a > 1 by A2,Th29;
then  (1/c) to_power b/(1/c) to_power a*(1/c) to_power a >
  1*(1/c) to_power a by A4,XREAL_1:68;
then  (1/c) to_power b > (1/c) to_power a by A5,XCMPLX_1:87;
then  1 to_power b/c to_power b > (1/c) to_power a by A2,Th31;
then  1 / c to_power b > (1/c) to_power a by Th26;
then  1 / c to_power b > 1 to_power a/c to_power a by A2,Th31;
then  1 / c to_power b > 1 / c to_power a by Th26;
  hence thesis by A6,XREAL_1:91;
end;
