reserve X for non empty set,
        x for Element of X,
        S for SigmaField of X,
        M for sigma_Measure of S,
        f,g,f1,g1 for PartFunc of X,REAL,
        l,m,n,n1,n2 for Nat,
        a,b,c for Real;
reserve k for positive Real;

theorem Th20:
for f be PartFunc of X,REAL, k be Real, E be set holds
  (f|E) to_power k = (f to_power k)|E
proof
   let f be PartFunc of X,REAL;
   let k be Real;
   let E be set;
A1:dom((f|E) to_power k) = dom(f|E) by MESFUN6C:def 4; then
   dom((f|E) to_power k) = dom f /\ E by RELAT_1:61; then
A2:dom((f|E) to_power k) = dom(f to_power k) /\ E by MESFUN6C:def 4; then
A3:dom((f|E) to_power k) = dom((f to_power k)|E) by RELAT_1:61;
   now let x be Element of X;
    assume A4: x in dom((f|E) to_power k); then
    ((f|E) to_power k).x = ((f|E).x) to_power k by MESFUN6C:def 4; then
A5: ((f|E) to_power k).x = (f.x) to_power k by A1,A4,FUNCT_1:47;
    x in dom(f to_power k) by A2,A4,XBOOLE_0:def 4; then
    ((f|E) to_power k).x = (f to_power k).x by A5,MESFUN6C:def 4;
    hence ((f|E) to_power k).x = ((f to_power k)|E).x by A4,A3,FUNCT_1:47;
   end;
   hence thesis by A3,PARTFUN1:5;
end;
