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

theorem Th30:
  for X being finite non empty Subset of REAL, f being Function of
  [:X,X:],REAL st f is symmetric nonnegative holds {X} in fam_class(f)
proof
  let X be finite non empty Subset of REAL, f be Function of [:X,X:],REAL such
  that
A1: f is symmetric nonnegative;
  dom f = [:X,X:] by FUNCT_2:def 1;
  then reconsider rn = rng f as finite non empty Subset of REAL by RELAT_1:42;
  reconsider A1 = max rn as Real;
  now
    set x = the Element of X;
    assume
A2: A1 is negative;
    f.(x,x) <= A1 by Th26;
    hence contradiction by A1,A2;
  end;
  then reconsider A19 = A1 as non negative Real;
  now
    let x be object;
    assume x in X;
    then reconsider x1 = x as Element of X;
    f.(x1,x1) <= A1 by Th26;
    hence [x,x] in low_toler(f,A1) by Def3;
  end;
  then low_toler(f,A19) is_reflexive_in X by RELAT_2:def 1;
  then reconsider R = low_toler(f,A19)[*] as Equivalence_Relation of X by A1
,Th22;
  Class R in fam_class(f) by Def5;
  hence thesis by Th27;
end;
