theorem
  0 <= k & E c= dom f & f is E-measurable implies |.f.| to_power k
  is E-measurable
proof
  assume that
A1: 0 <= k and
A2: E c= dom f and
A3: f is E-measurable;
A4: |.f.| is nonnegative by Lm1;
  E c= dom |.f.| by A2,VALUED_1:def 11;
  hence thesis by A1,A2,A3,A4,MESFUN6C:29,30;
end;
