reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  c for Complex,
  E,A,B for Element of S;

theorem Th27:
  for x be object,X,S,E for f be PartFunc of X,REAL st f is
  nonnegative holds (f.x) to_power (1/2) = sqrt(f.x)
proof
  let x be object,X,S,E;
  let f be PartFunc of X,REAL;
  assume f is nonnegative;
  then
A1: 0 <= f.x by MESFUNC6:51;
  hence (f.x) to_power (1/2) = 2-root (f.x) by POWER:45
    .= 2 -Root (f.x) by A1,POWER:def 1
    .= sqrt (f.x) by A1,PREPOWER:32;
end;
