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

theorem Th9:
  for X being non empty set, R being Relation of X st
  X c= field R  & R is_symmetric_in X holds R[*] is Equivalence_Relation of X
proof
  let X be non empty set, R be Relation of X such that
A1: X c= field R  and
A2: R is_symmetric_in X;
  R[*] is_reflexive_in X by A1,Th5;
  then
A3: dom(R[*]) = X & field(R[*]) = X by ORDERS_1:13;
  R[*] is_symmetric_in X & R[*] is_transitive_in X by A1,A2,Th7,Th8;
  hence thesis by A3,PARTFUN1:def 2,RELAT_2:def 11,def 16;
end;
