reserve i for Nat,
  j for Element of NAT,
  X,Y,x,y,z for set;

theorem Th32:
  for p,q being FinSequence st p^q is quasi-loci
  holds p is quasi-loci & q is FinSequence of Vars
proof
  let p,q be FinSequence;
  assume
A1: p^q is quasi-loci;
  then
A2: p is one-to-one FinSequence of Vars by FINSEQ_1:36,FINSEQ_3:91;
  now
    let i be Nat, x be variable such that
A3: i in dom p and
A4: x = p.i;
    let y be variable such that
A5: y in vars x;
A6: dom p c= dom (p^q) by FINSEQ_1:26;
    x = (p^q).i by A3,A4,FINSEQ_1:def 7;
    then consider j being Nat such that
A7: j in dom (p^q) and
A8: j < i and
A9: y = (p^q).j by A1,A3,A5,A6,Th30;
    take j;
A10: dom p = Seg len p by FINSEQ_1:def 3;
    dom (p^q) = Seg len (p^q) by FINSEQ_1:def 3;
    then
A11: j >= 1 by A7,FINSEQ_1:1;
    i <= len p by A3,A10,FINSEQ_1:1;
    then j < len p by A8,XXREAL_0:2;
    hence j in dom p & j < i by A8,A10,A11;
    hence y = p.j by A9,FINSEQ_1:def 7;
  end;
  hence thesis by A1,A2,Th30,FINSEQ_1:36;
end;
