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 Th6:
  for X being set, x,y being object, R being symmetric Relation of X
  st [x,y] in R holds [y,x] in R
proof
  let X be set, x,y be object, R be symmetric Relation of X;
  assume A1: [x,y] in R;
  then x in field R & y in field R by RELAT_1:15;
  hence thesis by A1, RELAT_2:def 3, RELAT_2:def 11;
end;
