reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;
reserve r,r1,r2,p for Real;
reserve f,f1,f2 for PartFunc of C,REAL;
reserve f for real-valued Function;

theorem
  (0(#)f)|Y is bounded
proof
  reconsider p1 = 0 as Real;
  set r = 0;
  now
    take p=p1;
    let c be object;
    assume c in Y /\ dom (r(#)f);
    then
A1: c in dom (r(#)f) by XBOOLE_0:def 4;
    r*f.c <= 0;
    hence (r(#)f).c <= p by A1,VALUED_1:def 5;
  end;
  hence (r(#)f)|Y is bounded_above by Th70;
  now
    take p=p1;
    let c be object;
    assume c in Y /\ dom (r(#)f);
    then
A2: c in dom (r(#)f) by XBOOLE_0:def 4;
    0 <= r*f.c;
    hence p <= (r(#)f).c by A2,VALUED_1:def 5;
  end;
  hence thesis by Th71;
end;
