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 Th19:
  for R being total symmetric Relation of X holds y in Class (R,x)
  iff [y,x] in R
proof
  let R be total symmetric Relation of X;
  thus y in Class(R,x) implies [y,x] in R
  proof
    assume y in Class(R,x);
    then ex z being object st [z,y] in R & z in {x} by RELAT_1:def 13;
    then [x,y] in R by TARSKI:def 1;
    hence thesis by Th6;
  end;
  assume [y,x] in R;
  then
A1: [x,y] in R by Th6;
  x in {x} by TARSKI:def 1;
  hence thesis by A1,RELAT_1:def 13;
end;
