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

theorem Th17:
  f is symmetric implies low_toler(f,a) is_symmetric_in X
proof
  assume
A1: f is symmetric;
  now
    let x,y be object such that
A2: x in X & y in X and
A3: [x,y] in low_toler(f,a);
    reconsider x1 = x, y1 = y as Element of X by A2;
    f.(x1,y1) <= a by A3,Def3;
    then f.(y1,x1) <= a by A1,METRIC_1:def 4;
    hence [y,x] in low_toler(f,a) by Def3;
  end;
  hence thesis by RELAT_2:def 3;
end;
