reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i for Integer;
reserve r for Real;
reserve p for Prime;

theorem Th9:
  <*i*> mod n = <*i mod n*>
  proof
    set f = <*i*>;
    set p = f mod n;
    dom p = dom f by EULER_2:def 1;
    then len p = len f by FINSEQ_3:29;
    then
A1: len p = 1 by FINSEQ_1:40;
A2: dom f = {1} by FINSEQ_1:2,def 8;
    1 in {1} by TARSKI:def 1;
    then p.1 = f.1 mod n by A2,EULER_2:def 1
    .= i mod n;
    hence thesis by A1,FINSEQ_1:40;
  end;
