reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;

theorem Th9:
  R is Equivalence_Relation of X iff
    R is reflexive symmetric transitive & field R = X
      by ORDERS_1:12,ORDERS_1:13,PARTFUN1:def 2;
