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;
reserve v,u for VECTOR of RLSp_LpFunct(M,k);
reserve v,u for VECTOR of RLSp_AlmostZeroLpFunct(M,k);
reserve x for Point of Pre-Lp-Space(M,k);
reserve x,y for Point of Lp-Space(M,k);

theorem Th69:
for F be Functional_Sequence of X,REAL st
  F is with_the_same_dom holds (abs F) is with_the_same_dom
proof
   let F be Functional_Sequence of X,REAL;
   assume A1: F is with_the_same_dom;
   for n,m be Nat holds dom((abs F).n) = dom((abs F).m)
   proof
    let n,m be Nat;
    (abs F).n = abs(F.n) & (abs F).m = abs(F.m) by SEQFUNC:def 4; then
    dom((abs F).n) = dom(F.n) & dom((abs F).m) = dom(F.m) by VALUED_1:def 11;
    hence dom((abs F).n) = dom((abs F).m) by A1,MESFUNC8:def 2;
   end;
   hence abs F is with_the_same_dom by MESFUNC8:def 2;
end;
