reserve x,X,Y for set;
reserve g,r,r1,r2,p,p1,p2 for Real;
reserve R for Subset of REAL;
reserve seq,seq1,seq2,seq3 for Real_Sequence;
reserve Ns for increasing sequence of NAT;
reserve n for Nat;
reserve W for non empty set;
reserve h,h1,h2 for PartFunc of W,REAL;

theorem
  for h being PartFunc of W,REAL, seq being sequence of W holds
  rng seq c= dom (h|X) implies abs((h|X)/*seq) = ((abs(h))|X)/*seq
proof
  let h be PartFunc of W,REAL, seq be sequence of W;
  assume
A1: rng seq c= dom (h|X);
A2: dom (h|X) = dom h /\ X by RELAT_1:61
    .= dom abs(h) /\ X by VALUED_1:def 11
    .= dom (abs(h)|X) by RELAT_1:61;
  now
    let n be Element of NAT;
A3: seq.n in rng seq by VALUED_0:28;
    then seq.n in dom (h|X) by A1;
    then seq.n in dom h /\ X by RELAT_1:61;
    then
A4: seq.n in X by XBOOLE_0:def 4;
    thus (abs((h|X)/*seq)).n = |. ((h|X)/*seq).n .| by SEQ_1:12
      .= |. (h|X).(seq.n) .| by A1,FUNCT_2:108
      .= |. h.(seq.n) .| by A1,A3,FUNCT_1:47
      .= (abs(h)).(seq.n) by VALUED_1:18
      .= ((abs(h))|X).(seq.n) by A4,FUNCT_1:49
      .= (((abs(h))|X)/*seq).n by A1,A2,FUNCT_2:108;
  end;
  hence thesis by FUNCT_2:63;
end;
