
theorem PowerIn01:
  for a,b being Element of [.0,1.] st a > 0 or b > 0 holds
    b to_power a in [.0,1.]
  proof
    let a,b be Element of [.0,1.];
YY: b <= 1 & b >= 0 by XXREAL_1:1;
XX: a >= 0 by XXREAL_1:1;
    assume a > 0 or b > 0; then
    per cases;
    suppose
S1:   a > 0 & b <> 0 & b <> 1; then
B1:   a > 0 & b > 0 by XXREAL_1:1;
ZZ:   b < 1 by S1,XXREAL_0:1,YY;
      b to_power a < b to_power 0 by POWER:40,B1,ZZ; then
A1:   b to_power a <= 1 by POWER:24;
      b to_power a > 0 by B1,POWER:34;
      hence thesis by A1,XXREAL_1:1;
    end;
    suppose
S1:   a > 0 & b <> 0 & b = 1; then
A1:   b to_power a <= 1 by POWER:26;
      b to_power a > 0 by S1,POWER:34;
      hence thesis by A1,XXREAL_1:1;
    end;
    suppose
B1:   a > 0 & b = 0; then
A1:   b to_power a <= 1 by POWER:def 2;
      b to_power a >= 0 by POWER:def 2,B1;
      hence thesis by A1,XXREAL_1:1;
    end;
    suppose B1: b > 0 & b <> 1;
      b <= 1 by XXREAL_1:1; then
      b to_power a <= 1 to_power a by XX,B1,HOLDER_1:3; then
A1:   b to_power a <= 1 by POWER:26;
      b to_power a > 0 by B1,POWER:34;
      hence thesis by A1,XXREAL_1:1;
    end;
    suppose B1: b > 0 & b = 1; then
A1:   b to_power a <= 1 by POWER:26;
      b to_power a > 0 by B1,POWER:34;
      hence thesis by A1,XXREAL_1:1;
    end;
  end;
