reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,j,k,m,n for Nat,
  p,q,r for Element of CQC-WFF(Al),
  x,y,y0 for bound_QC-variable of Al,
  X for Subset of CQC-WFF(Al),
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  Sub for CQC_Substitution of Al,
  f,f1,g,h,h1 for FinSequence of CQC-WFF(Al);

theorem Th1:
  f is_Subsequence_of g implies rng f c= rng g &
  ex N being Subset of NAT st rng f c= rng (g|N)
proof
  assume f is_Subsequence_of g;
  then consider N being Subset of NAT such that
A1: f c= Seq (g|N);
A2: rng (g|N) c= rng g by RELAT_1:70;
A3: now
    rng (Seq (g|N)) = rng ((g|N) * Sgm (dom (g|N))) by FINSEQ_1:def 15;
    then
A4: rng (Seq (g|N)) c= rng (g|N) by RELAT_1:26;
    let a be object;
    assume a in rng f;
    then consider n being Nat such that
A5: n in dom f and
A6: f.n = a by FINSEQ_2:10;
    [n,f.n] in f by A5,FUNCT_1:1;
    then
A7: (Seq (g|N)).n = a by A1,A6,FUNCT_1:1;
    dom f c= dom Seq (g|N) by A1,RELAT_1:11;
    then a in rng (Seq (g|N)) by A5,A7,FUNCT_1:3;
    hence a in rng (g|N) by A4;
  end;
  then rng f c= rng (g|N);
  hence rng f c= rng g by A2;
  take N;
  thus thesis by A3;
end;
