reserve n,n1,m,m1,k for Nat;
reserve x,X,X1 for set;
reserve g,g1,g2,t,x0,x1,x2 for Complex;
reserve s1,s2,q1,seq,seq1,seq2,seq3 for Complex_Sequence;
reserve Y for Subset of COMPLEX;
reserve f,f1,f2,h,h1,h2 for PartFunc of COMPLEX,COMPLEX;
reserve p,r,s for Real;
reserve Ns,Nseq for increasing sequence of NAT;

theorem Th10:
  rng seq c= dom (h^) implies h/*seq is non-zero
proof
  assume
A1: rng seq c= dom (h^);
  then
A2: dom h \ h"{0c} c= dom h & rng seq c= dom h \ h"{0c} by CFUNCT_1:def 2
,XBOOLE_1:36;
  now
    given n being Element of NAT such that
A3: (h/*seq).n=0c;
    seq.n in rng seq by VALUED_0:28;
    then seq.n in dom (h^) by A1;
    then seq.n in dom h \ h"{0c} by CFUNCT_1:def 2;
    then seq.n in dom h & not seq.n in h"{0c} by XBOOLE_0:def 5;
    then
A4: not h/.(seq.n) in {0c} by PARTFUN2:26;
    h/.(seq.n) =0c by A2,A3,FUNCT_2:109,XBOOLE_1:1;
    hence contradiction by A4,TARSKI:def 1;
  end;
  hence thesis by COMSEQ_1:4;
end;
