reserve A,B,a,b,c,d,e,f,g,h for set;

theorem
  for R being reflexive antisymmetric RelStr, S being RelStr holds (ex f
  being Function of R,S st for x,y being Element of R holds [x,y] in the
  InternalRel of R iff [f.x,f.y] in the InternalRel of S) iff S embeds R
proof
  let R being reflexive antisymmetric RelStr, S being RelStr;
A1: now
    assume ex f being Function of R,S st for x,y being Element of R holds [x,
    y] in the InternalRel of R iff [f.x,f.y] in the InternalRel of S;
    then consider f being Function of R,S such that
A2: for x,y being Element of R holds [x,y] in the InternalRel of R iff
    [f.x,f.y] in the InternalRel of S;
    for x1,x2 being object st x1 in dom f & x2 in dom f & f.x1 = f.x2 holds
    x1 = x2
    proof
      let x1,x2 be object;
      assume that
A3:   x1 in dom f and
A4:   x2 in dom f and
A5:   f.x1 = f.x2;
      reconsider x1,x2 as Element of R by A3,A4;
A6:   the InternalRel of R is_reflexive_in the carrier of R by ORDERS_2:def 2;
      then [x2,x2] in the InternalRel of R by A3;
      then [f.x2,f.x1] in the InternalRel of S by A2,A5;
      then [x2,x1] in the InternalRel of R by A2;
      then
A7:   x2 <= x1 by ORDERS_2:def 5;
      [x1,x1] in the InternalRel of R by A3,A6;
      then [f.x1,f.x2] in the InternalRel of S by A2,A5;
      then [x1,x2] in the InternalRel of R by A2;
      then x1 <= x2 by ORDERS_2:def 5;
      hence thesis by A7,ORDERS_2:2;
    end;
    then f is one-to-one by FUNCT_1:def 4;
    hence S embeds R by A2;
  end;
  thus thesis by A1;
end;
