
theorem Th51:
  for p being Prime, n being non zero Element of NAT holds
  p |-count <*n*> = <* (p |-count n) *>
proof
  let p be Prime;
  let n be non zero Element of NAT;
A1: dom (p |-count <*n*>) = Seg len (p |-count <*n*>) by FINSEQ_1:def 3
    .= Seg len <*n*> by Def1
    .= Seg 1 by FINSEQ_1:39;
A2: for k being Nat st k in dom (p |-count <*n*>) holds (p |-count <*n*>).k
  = <* p |-count n *>.k
  proof
    let k be Nat;
    assume
A3: k in dom (p |-count <*n*>);
    then 1<=k & k<=1 by A1,FINSEQ_1:1;
    then
A4: k=1 by XXREAL_0:1;
    (p |-count <*n*>).k = p |-count <*n*>.k by A3,Def1;
    then (p |-count <*n*>).k = p |-count n by A4;
    hence thesis by A4;
  end;
  Seg 1 = Seg len <* p |-count n *> by FINSEQ_1:39;
  then dom (p |-count <*n*>) = dom <* (p |-count n) *> by A1,FINSEQ_1:def 3;
  hence thesis by A2,FINSEQ_1:13;
end;
