reserve A,X for non empty set;
reserve f for PartFunc of [:X,X:],REAL;
reserve a for Real;

theorem Th19:
  for X being non empty set, f being PartFunc of [:X,X:], REAL,
      a1,a2 being Real st a1 <= a2 holds low_toler(f,a1) c= low_toler(f,a2)
proof
  let X be non empty set, f be PartFunc of [:X,X:], REAL, a1,a2 be Real
  such that
A1: a1 <= a2;
    let p be object such that
A2: p in low_toler(f,a1);
    consider x,y being object such that
A3: x in X & y in X and
A4: p = [x,y] by A2,ZFMISC_1:def 2;
    reconsider x1 = x, y1 = y as Element of X by A3;
    f.(x1,y1) <= a1 by A2,A4,Def3;
    then f.(x1,y1) <= a2 by A1,XXREAL_0:2;
    hence p in low_toler(f,a2) by A4,Def3;
end;
