theorem
  R is antisymmetric implies R is_antisymmetric_in S
proof
  assume R is antisymmetric; then
A1: R is_antisymmetric_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;
