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

theorem Th31:
  for X being non empty set, f being PartFunc of [:X,X:],REAL
  holds fam_class(f) is Classification of X
proof
  let X be non empty set,f be PartFunc of [:X,X:],REAL;
  for A,B being a_partition of X st A in fam_class(f) & B in fam_class(f)
  holds A is_finer_than B or B is_finer_than A
  proof
    let A,B be a_partition of X;
    assume that
A1: A in fam_class(f) and
A2: B in fam_class(f);
    consider a1 being non negative Real, R1 being
    Equivalence_Relation of X such that
A3: R1 = low_toler(f,a1)[*] and
A4: Class R1 = A by A1,Def5;
    consider a2 being non negative Real, R2 being
    Equivalence_Relation of X such that
A5: R2 = low_toler(f,a2)[*] and
A6: Class R2 = B by A2,Def5;
    now
      per cases;
      suppose
A7:     a1 <= a2;
        now
          let x be set;
          assume x in A;
          then consider c being object such that
A8:       c in X and
A9:       x = Class(R1,c) by A4,EQREL_1:def 3;
          consider y being set such that
A10:      y = Class(R2,c);
          take y;
          thus y in B by A6,A8,A10,EQREL_1:def 3;
          thus x c= y by A3,A5,A7,A9,A10,Lm2;
        end;
        hence thesis;
      end;
      suppose
A11:    a1 > a2;
        now
          let y be set;
          assume y in B;
          then consider c being object such that
A12:      c in X and
A13:      y = Class(R2,c) by A6,EQREL_1:def 3;
          consider x being set such that
A14:      x = Class(R1,c);
          take x;
          thus x in A by A4,A12,A14,EQREL_1:def 3;
          thus y c= x by A3,A5,A11,A13,A14,Lm2;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  hence thesis by Def1;
end;
