reserve X for set;
reserve n,i for Element of NAT;
reserve a,b,c,d,e,r,x0 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve f,g,h for PartFunc of REAL,REAL n;
reserve E for Element of REAL n;

theorem
  f is bounded & A c= dom f implies f|A is bounded
  proof
    assume
A1: f is bounded & A c= dom f;
    let i; set P = proj(i,n);
    assume i in Seg n;
    then P*f is bounded by A1;
    then consider r be Real such that
A2: for c be object st c in dom (P*f) holds
    |.((P*f)).c.|<=r by RFUNCT_1:72;
    now let c be object;
      assume c in A /\ dom (P*f);
      then c in dom (P*f) by XBOOLE_0:def 4;
      hence |.((P*f)).c.|<=r by A2;
    end;
    then (P*f) |A is bounded by RFUNCT_1:73;
    hence (P*(f|A)) is bounded by RELAT_1:83;
  end;
