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;
reserve P for Permutation of dom f;

theorem Th31:
  for c being set st 1 <= n holds rng IdFinS(c,n) = rng <*c*>
proof
  let c be set such that
A1: 1 <= n;
  n in Seg n by A1,FINSEQ_1:1;
  then
A2: IdFinS(c,n).n = c by FUNCOP_1:7;
  thus rng IdFinS(c,n) c= rng <*c*>
  proof
    let a be object;
    assume a in rng IdFinS(c,n);
    then consider i being Nat such that
A3: i in dom IdFinS(c,n) and
A4: IdFinS(c,n).i = a by FINSEQ_2:10;
    i in Seg len IdFinS(c,n) by A3,FINSEQ_1:def 3;
    then i in Seg n by CARD_1:def 7;
    then a = c by A4,FUNCOP_1:7;
    then a in {c} by TARSKI:def 1;
    hence thesis by FINSEQ_1:38;
  end;
  let a be object;
  assume a in rng <*c*>;
  then a in {c} by FINSEQ_1:38;
  then
A5: a = c by TARSKI:def 1;
  n = len IdFinS(c,n) by CARD_1:def 7;
  then n in dom IdFinS(c,n) by A1,FINSEQ_3:25;
  hence thesis by A5,A2,FUNCT_1:def 3;
end;
