theorem
  x is_inferior_of R & R is antisymmetric implies x is_minimal_in R
proof
  assume that
A1: x is_inferior_of R and
A2: R is antisymmetric;
A3: R is_antisymmetric_in field R by A2;
  thus
A4: x in field R by A1;
  let y;
  assume that
A5: y in field R and
A6: y <> x and
A7: [y,x] in R;
  [x,y] in R by A1,A5,A6;
  hence thesis by A4,A5,A6,A7,A3;
end;
