theorem Th23:
  for k being Nat st
  len p + 1 <= k & k <= len p + len q holds (p^q).k=q.(k-len p)
proof
  let k be Nat;
  assume that
A1: len p + 1 <= k and
A2: k <= len p + len q;
  consider m be Nat such that
A3: (len p + 1)+m = k by A1,NAT_1:10;
A4: len p+(1+m) = k by A3;
  1+m = k - len p by A3;
  then
A5: 1 <= 1+m by A1,XREAL_1:19;
  k-len p <= len q by A2,XREAL_1:20;
  then 1+m in Seg len q by A3,A5;
  then 1+m in dom q by Def3;
  hence thesis by A4,Def7;
end;
