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 is non-zero
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
    given n such that
A3: (h/*seq).n=0;
A4: n in NAT by ORDINAL1:def 12;
    seq.n in rng seq by VALUED_0:28;
    then seq.n in dom (h^) by A1;
    then seq.n in dom h \ h"{0} by RFUNCT_1:def 2;
    then seq.n in dom h & not seq.n in h"{0} by XBOOLE_0:def 5;
    then
A5: not h.(seq.n) in {0} by FUNCT_1:def 7;
    h.(seq.n) =0 by A2,A3,FUNCT_2:108,XBOOLE_1:1,A4;
    hence contradiction by A5,TARSKI:def 1;
  end;
  hence thesis by SEQ_1:5;
end;
