theorem Th12:
for k be positive Real,
  X be non empty set holds (X -->0) to_power k = X --> 0
proof
   let k be positive Real, X be non empty set;
A1:dom ((X -->0) to_power k) = dom (X-->0) by MESFUN6C:def 4;
   now let x be Element of X;
    assume x in dom((X -->0) to_power k); then
    ((X -->0) to_power k).x = ((X -->0).x) to_power k by MESFUN6C:def 4; then
    ((X -->0) to_power k).x = 0 to_power k by FUNCOP_1:7; then
    ((X -->0) to_power k).x = 0 by POWER:def 2;
    hence ((X -->0) to_power k).x = (X --> 0).x by FUNCOP_1:7;
   end;
   hence thesis by A1,PARTFUN1:5;
end;
