reserve a, b, n for Nat,
  r for Real,
  f for FinSequence of REAL;
reserve p for Prime;

theorem Th13:
  (f^<*r*>) |^ a = (f|^a)^(<*r*>|^a)
proof
A1: len (f|^a) = len f by Def1;
A2: len ((f^<*r*>) |^ a) = len (f^<*r*>) by Def1
    .= len f + len <*r*> by FINSEQ_1:22
    .= len f + 1 by FINSEQ_1:40;
  then
A3: dom ((f^<*r*>) |^ a) = Seg(len f +1) by FINSEQ_1:def 3;
A4: now
    let i be Nat such that
A5: i in dom ((f^<*r*>) |^ a);
A6: 1 <= i by A3,A5,FINSEQ_1:1;
A7: i <= len f + 1 by A3,A5,FINSEQ_1:1;
    per cases by A7,XXREAL_0:1;
    suppose
      i < len f +1;
      then
A8:   i <= len f by NAT_1:13;
      then
A9:   i in dom f by A6,FINSEQ_3:25;
A10:  i in dom (f|^a) by A1,A6,A8,FINSEQ_3:25;
      thus ((f^<*r*>) |^ a).i = (f^<*r*>).i |^ a by A5,Def1
        .= f.i |^a by A9,FINSEQ_1:def 7
        .= (f|^a).i by A10,Def1
        .= ((f|^a)^(<*r*>|^a)).i by A10,FINSEQ_1:def 7;
    end;
    suppose
A11:  i = len f +1;
      thus ((f^<*r*>) |^ a).i = (f^<*r*>).i |^ a by A5,Def1
        .= r |^ a by A11,FINSEQ_1:42
        .= ((f|^a)^(<*r|^a*>)).i by A1,A11,FINSEQ_1:42
        .= ((f|^a)^(<*r*>|^a)).i by Th12;
    end;
  end;
  len ((f|^a)^(<*r*>|^a)) = len (f|^a) + len (<*r*>|^a) by FINSEQ_1:22
    .= len f + len (<*r*>|^a) by Def1
    .= len f + len (<*r|^a*>) by Th12
    .= len f + 1 by FINSEQ_1:40;
  hence thesis by A2,A4,FINSEQ_2:9;
end;
