reserve x for set;
reserve k, l for Nat;
reserve p, q for FinSequence;

theorem Th2:
  k > len p & k <= len (p^q) implies ex l st k = len p + l & l >= 1
  & l <= len q
proof
A1: 0 + 1 = 1;
  assume that
A2: k > len p and
A3: k <= len (p^q);
  consider l such that
A4: k = len p + l by A2,NAT_1:10;
  take l;
  thus k = len p + l by A4;
  len p + l > len p + 0 by A2,A4;
  then l > 0;
  hence l >= 1 by A1,NAT_1:13;
  len p + l <= len p + len q by A3,A4,FINSEQ_1:22;
  hence thesis by XREAL_1:6;
end;
