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 Th13:
for f be PartFunc of X,REAL, D be set holds abs(f|D) = (abs f)|D
proof
   let f be PartFunc of X,REAL;
   let D be set;
A1:dom abs(f|D) = dom (f|D) by VALUED_1:def 11; then
   dom abs(f|D) = dom f /\ D by RELAT_1:61; then
   dom abs(f|D) = dom abs f /\ D by VALUED_1:def 11; then
A2:dom abs(f|D) = dom((abs f)|D) by RELAT_1:61;
   for x be Element of X st x in dom abs(f|D) holds
     (abs(f|D)).x = ((abs f)|D).x
   proof
    let x be Element of X;
    assume A3: x in dom abs(f|D); then
    x in dom f by A1,RELAT_1:57; then
A4: x in dom abs f by VALUED_1:def 11;
    (abs(f|D)).x = |.(f|D).x.| by A3,VALUED_1:def 11; then
    (abs(f|D)).x = |.f.x.| by A3,A1,FUNCT_1:47; then
    (abs(f|D)).x = (abs f).x by A4,VALUED_1:def 11;
    hence (abs(f|D)).x = ((abs f)|D).x by A3,A2,FUNCT_1:47;
   end;
   hence thesis by A2,PARTFUN1:5;
end;
