
theorem Th42:
  for A being Order holds
    EqRelOf A = id the carrier of A
proof
  let A be Order;
  per cases;
  suppose A is empty;
    hence thesis;
  end;
  suppose A is non empty;
    then reconsider B = A as non empty Order;
    defpred P[set, set] means $1 = $2;
    A1: for x, y being Element of B holds [x,y] in EqRelOf B iff P[x,y]
    proof
      let x, y be Element of B;
      hereby
        assume [x,y] in EqRelOf B;
        then x <= y & y <= x by Def6;
        hence P[x,y] by ORDERS_2:2;
      end;
      assume P[x,y];
      then x <= y & y <= x;
      hence [x,y] in EqRelOf B by Def6;
    end;
    A2: for x, y being Element of B holds
      [x,y] in id the carrier of B iff P[x,y] by RELAT_1:def 10;
    thus EqRelOf A = EqRelOf B
      .= id the carrier of B from RELSET_1:sch 4(A1, A2)
      .= id the carrier of A;
  end;
end;
