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^) implies (h^)/*seq =(h/*seq)"
proof
  let h be PartFunc of W,REAL, seq be sequence of W;
  assume
A1: rng seq c= dom (h^);
  then
A2: dom h \ h"{0} c= dom h & rng seq c= dom h \ h"{0} by RFUNCT_1:def 2
,XBOOLE_1:36;
  now
    let n be Element of NAT;
A3: seq.n in rng seq by VALUED_0:28;
    thus ((h^)/*seq).n = (h^).(seq.n) by A1,FUNCT_2:108
      .= (h.(seq.n))" by A1,A3,RFUNCT_1:def 2
      .= ((h/*seq).n)" by A2,FUNCT_2:108,XBOOLE_1:1
      .= ((h/*seq)").n by VALUED_1:10;
  end;
  hence thesis by FUNCT_2:63;
end;
