reserve X,Y,x,y for set;
reserve A for non empty Poset;
reserve a,a1,a2,a3,b,c for Element of A;
reserve S,T for Subset of A;
reserve f for Choice_Function of BOOL(the carrier of A);
reserve fC,fC1,fC2 for Chain of f;
reserve R for Relation,
  A for non empty Poset,
  C for Chain of A,
  S for Subset of A,
  a,a1,a2,b,c1,c2 for Element of A;

theorem
  a is_minimal_in the InternalRel of A iff for b holds not b < a
proof
A1: the carrier of A = field(the InternalRel of A) by ORDERS_1:15;
  thus a is_minimal_in the InternalRel of A implies for b holds not b < a
  proof
    assume
A2: a is_minimal_in the InternalRel of A;
    let b;
    a = b or not [b,a] in the InternalRel of A by A1,A2;
    then a = b or not b <= a;
    hence thesis;
  end;
  assume
A3: for b holds not b < a;
  thus a in field(the InternalRel of A) by A1;
  let y;
  assume that
A4: y in field(the InternalRel of A) and
A5: y <> a and
A6: [y,a] in the InternalRel of A;
  reconsider b = y as Element of A by A4,ORDERS_1:15;
  b <= a by A6;
  then b < a by A5;
  hence thesis by A3;
end;
