
theorem Th33:
  for p being non empty FinSequence, q being FinSequence for i
  being Nat st i < len q holds (p$^q).(len p+i) = q.(i+1)
proof
  let p be non empty FinSequence, q be FinSequence;
  let i be Nat;
A1: i+1 >= 1 by NAT_1:11;
  assume
A2: i < len q;
  then consider j being Nat, r being FinSequence such that
A3: len p = j+1 and
A4: r = p|Seg j and
A5: p$^q = r^q by CARD_1:27,REWRITE1:def 1;
  i+1 <= len q by A2,NAT_1:13;
  then
A6: i+1 in dom q by A1,FINSEQ_3:25;
  j < len p by A3,NAT_1:13;
  then len r = j by A4,FINSEQ_1:17;
  then len p+i =len r+(i+1) by A3;
  hence thesis by A5,A6,FINSEQ_1:def 7;
end;
