reserve X,Y for set,
  x,x1,x2,y,y1,y2,z for set,
  f,g,h for Function;
reserve M for non empty set;
reserve D for non empty set;
reserve P for Relation;
reserve O for Order of X;
reserve R,P for Relation,
  X,X1,X2,Y,Z,x,y,z,u for set,
  g,h for Function,
  O for Order of X,
  D for non empty set,
  d,d1,d2 for Element of D,
  A1,A2,B for Ordinal,
  L,L1,L2 for Sequence;

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;
