
theorem Th5:
  for p,q being FinSequence, k being Element of NAT holds len p + 1
  <= k implies (p^q).k=q.(k-len p)
proof
  let p,q be FinSequence;
  let k be Element of NAT;
  assume
A1: len p + 1 <= k;
  per cases;
  suppose
    k <= len p + len q;
    hence thesis by A1,FINSEQ_1:23;
  end;
  suppose
A2: not k <= len p + len q;
    then k-len p > len q by XREAL_1:20;
    then
A3: not k-len p in dom q by FINSEQ_3:25;
    not k <= len (p^q) by A2,FINSEQ_1:22;
    then not k in dom (p^q) by FINSEQ_3:25;
    hence (p^q).k= {} by FUNCT_1:def 2
      .= q.(k-len p) by A3,FUNCT_1:def 2;
  end;
end;
