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:];

theorem Th4:
  nabla X is Equivalence_Relation of X
proof
  for x,y holds x in X & y in X & [x,y] in nabla X implies [y,x] in nabla
  X by ZFMISC_1:88;
  then
A1: nabla X is_symmetric_in X;
  for x,y,z st x in X & y in X & z in X & [x,y] in nabla X & [y,z] in
  nabla X holds [x,z] in nabla X by ZFMISC_1:87;
  then
A2: nabla X is_transitive_in X;
  field nabla X = X by ORDERS_1:12;
  hence thesis by A1,A2,RELAT_2:def 11,def 16;
end;
