reserve Y for non empty set,
  a for Function of Y,BOOLEAN,
  G for Subset of PARTITIONS(Y),
  P,Q for a_partition of Y;
reserve x,y,z for set,
  S, X for non empty set,
  R for Relation of X;

theorem
  R is symmetric implies R is_symmetric_in S
proof
  assume R is symmetric;
  then
A1: R is_symmetric_in field R;
  let x,y be object;
  assume x in S & y in S;
  assume
A2: [x,y] in R;
  then x in field R & y in field R by RELAT_1:15;
  hence thesis by A1,A2;
end;
