theorem Th97:
  for f being vf-sequence of t, B being FinSequence of the carrier of S
  st B = pr2 f holds pr1 f is B-sorts FinSequence of Union X
  proof
    let f be vf-sequence of t;
    let B be FinSequence of the carrier of S;
    assume Z0: B = pr2 f;
    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;
    pr1 f is FinSequence of Union X
    proof
      let a; assume a in rng pr1 f;
      then consider b such that
A2:   b in dom pr1 f & a = (pr1 f).b by FUNCT_1:def 3;
      reconsider b as Nat by A2;
A3:   dom pr1 f = dom f by MCART_1:def 12;
      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)`1 by A2,A3,MCART_1:def 12 .= [x,s]`1 by A1,A2,A3,A4,A5;
      hence a in Union X;
    end;
    then reconsider V = pr1 f as FinSequence of Union X;
    V is B-sorts
    proof
A6:   dom V = dom f = dom B by Z0,MCART_1:def 12,def 13;
      hence dom V = dom B;
      let i; assume
A7:   i in dom B;
      then g.i in rng g by A1,A6,FUNCT_1:def 3;
      then consider xi being Element of dom t such that
A4:   g.i = xi & ex s,x st t.xi = [x,s] by A1;
      consider s,x such that
A5:   t.xi = [x,s] by A4;
      B.i = (f.i)`2 & V.i = (f.i)`1 & f.i = [x,s]
      by Z0,A1,A4,A5,A6,A7,MCART_1:def 12,def 13;
      hence V.i in X.(B.i);
    end;
    hence pr1 f is B-sorts FinSequence of Union X;
  end;
