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 Th14:
for f be PartFunc of X,REAL st f is nonnegative holds abs f = f
proof
   let f be PartFunc of X,REAL;
A1:dom f = dom (abs f) by VALUED_1:def 11;
   assume A2: f is nonnegative;
   now let x be Element of X;
A3: f.x >= 0 by A2,MESFUNC6:51;
    assume x in dom f; then
    x in dom abs f by VALUED_1:def 11; then
    (abs f).x = |.f.x.| by VALUED_1:def 11;
    hence (abs f).x = f.x by A3,ABSVALUE:def 1;
   end;
   hence thesis by A1,PARTFUN1:5;
end;
