reserve r,p,x for Real;
reserve n for Element of NAT;
reserve A for non empty closed_interval Subset of REAL;
reserve Z for open Subset of REAL;

theorem
  for f being PartFunc of REAL,REAL,
  A being non empty closed_interval Subset of
REAL st (f(#)f)||A is total & (f(#)f)||A|A is bounded & (for x st x in A holds
  (f(#)f)||A.x >= 0) holds 0 <= ||..f,A..||
proof
  let f be PartFunc of REAL,REAL;
  let A be non empty closed_interval Subset of REAL;
  assume
A1: (f(#)f)||A is total;
  assume (f(#)f)||A|A is bounded & for x st x in A holds (f(#)f)||A.x >= 0;
  then |||(f,f,A)||| >= 0 by A1,INTEGRA2:32;
  hence thesis by SQUARE_1:def 2;
end;
