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,g be 1-sorted holds Carr {f,g} = {the carrier of f, the carrier of g }
proof
  let f,g be 1-sorted;
  thus Carr {f,g} c= {the carrier of f, the carrier of g}
  proof
    let a be object;
    assume a in Carr {f,g};
    then consider s be 1-sorted such that
A1: s in {f,g} and
A2: a = the carrier of s by Def7;
    per cases by A1,TARSKI:def 2;
    suppose
      s = f;
      hence thesis by A2,TARSKI:def 2;
    end;
    suppose
      s = g;
      hence thesis by A2,TARSKI:def 2;
    end;
  end;
  thus {the carrier of f, the carrier of g} c= Carr {f,g}
  proof
    let a be object;
A3: f in {f,g} by TARSKI:def 2;
A4: g in {f,g} by TARSKI:def 2;
    assume
A5: a in {the carrier of f, the carrier of g};
    per cases by A5,TARSKI:def 2;
    suppose
      a = the carrier of f;
      hence thesis by A3,Def7;
    end;
    suppose
      a = the carrier of g;
      hence thesis by A4,Def7;
    end;
  end;
end;
