reserve i,j for Nat;
reserve i,j for Nat,
  x for variable,
  l for quasi-loci;
reserve C for initialized ConstructorSignature,
  c for constructor OperSymbol of C;

theorem Th30:
  ex l st len l = i
  proof
    defpred P[Nat] means ex l st len l = $1;
    <*>Vars is quasi-loci & len <*>Vars = 0 by ABCMIZ_1:29; then
A1: P[ 0 ];
A2: P[j] implies P[j+1] proof given l such that
A3:   len l = j;
      reconsider l1 = l^<*[rng l, 1]*> as quasi-loci by Th29;
      take l1; thus thesis by A3,FINSEQ_2:16;
    end;
    P[j] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
