theorem Th96:
  for f being vf-sequence of t holds pr2 f is FinSequence of the carrier of S
  proof
    let f be vf-sequence of t;
    consider g being one-to-one FinSequence such that
A1: rng g = {xi where xi is Element of dom t: ex s,x st t.xi = [x,s]} &
    dom f = dom g & for i st i in dom f holds f.i = t.(g.i) by VFS;
    let a; assume a in rng pr2 f;
    then consider b such that
A2: b in dom pr2 f & a = (pr2 f).b by FUNCT_1:def 3;
    reconsider b as Nat by A2;
A3: dom pr2 f = dom f by MCART_1:def 13;
    then g.b in rng g by A1,A2,FUNCT_1:def 3;
    then consider xi being Element of dom t such that
A4: g.b = xi & ex s,x st t.xi = [x,s] by A1;
    consider s,x such that
A5: t.xi = [x,s] by A4;
    a = (f.b)`2 by A2,A3,MCART_1:def 13 .= [x,s]`2 by A1,A2,A3,A4,A5 .= s;
    hence a in the carrier of S;
  end;
