reserve T for non empty RelStr,
  a for Element of T;
reserve a for set;
reserve P for non empty POSet_set,
  A,B for Element of P;

theorem
  for f be 1-sorted holds Carr {f} = {the carrier of f}
proof
  let f be 1-sorted;
  thus Carr {f} c= {the carrier of f}
  proof
    let a be object;
    assume a in Carr {f};
    then consider s be 1-sorted such that
A1: s in {f} and
A2: a = the carrier of s by Def7;
    s = f by A1,TARSKI:def 1;
    hence thesis by A2,TARSKI:def 1;
  end;
A3: f in {f} by TARSKI:def 1;
  thus {the carrier of f} c= Carr {f}
  proof
    let a be object;
    assume a in {the carrier of f};
    then a = the carrier of f by TARSKI:def 1;
    hence thesis by A3,Def7;
  end;
end;
