reserve Al for QC-alphabet;
reserve p,q,p1,p2,q1 for Element of CQC-WFF(Al),
  k for Element of NAT,
  f,f1,f2,g for FinSequence of CQC-WFF(Al),
  a,b,b1,b2,c,i,n for Nat;

theorem Th18:
  f is_Subsequence_of g^f
proof
A1: for i be Nat st i in dom Seq((g^f)|seq(len g,len f)) holds Seq((g^f)|seq
  (len g,len f)).i = f.i
  proof
    let i be Nat;
    assume i in dom Seq((g^f)|seq(len g,len f));
    then
A2: i in dom f by Th17;
    then
A3: i in dom Sgm(seq(len g,len f)) by Th11;
    Seq((g^f)|seq(len g,len f)).i = (Sgm(seq(len g,len f)) * (g^f)).i by Th16;
    then Seq((g^f)|seq(len g,len f)).i = (g^f).(Sgm(seq(len g,len f)).i) by A3,
FUNCT_1:13;
    then Seq((g^f)|seq(len g,len f)).i = (g^f).(len g+i) by A3,Th13;
    hence thesis by A2,FINSEQ_1:def 7;
  end;
  dom Seq((g^f)|seq(len g,len f)) = dom f by Th17;
  then Seq((g^f)|seq(len g,len f)) = f by A1,FINSEQ_1:13;
  hence thesis by CALCUL_1:def 4;
end;
