reserve p,q,r for FinSequence,
  x,y for object;

theorem Th2:
  q <> {} implies (p^<*x*>)$^q = p^q
proof
  len <*x*> = 1 by FINSEQ_1:40;
  then
A1: len (p^<*x*>) = len p + 1 by FINSEQ_1:22;
  assume q <> {};
  then consider i being Nat, r being FinSequence such that
A2: len (p^<*x*>) = i+1 and
A3: r = (p^<*x*>)|Seg i and
A4: (p^<*x*>)$^q = r^q by Def1;
  i <= i+1 by NAT_1:11;
  then
A5: len r = i by A2,A3,FINSEQ_1:17;
  ex s being FinSequence st p^<*x*> = r^s by A3,FINSEQ_1:80;
  then consider t being FinSequence such that
A6: p^t = r by A2,A1,A5,FINSEQ_1:47;
  len r + 0 = len p + len t by A6,FINSEQ_1:22;
  then t = {} by A2,A1,A5;
  hence thesis by A4,A6,FINSEQ_1:34;
end;
