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;
