
theorem Th24:
  for K1 be Subfield of F_Rat holds INT c= the carrier of K1
  proof
    let K1 be Subfield of F_Rat, m be object;
    assume m in INT;
    then reconsider m as Integer;
    set C1 = the carrier of K1;
A1: NAT c= C1 by Th23;
    per cases;
    suppose 0 <= m;
      hence thesis by A1,INT_1:3;
    end;
    suppose 0 > m;
      then reconsider mm = -m as Element of NAT by INT_1:3;
      reconsider mmm = mm as Element of K1 by A1;
      consider mm1 be Element of K1 such that
      A2: mmm+mm1 =0.K1 by ALGSTR_0:def 11;
      A3: C1 c= C by EC_PF_1:def 1;
      then reconsider mm2 = mm1 as Element of K;
      reconsider mm3 = mm as Element of K by A2,A3;
      mm3+mm2 = 0.K1 by A2,Th17
      .= 0.K by EC_PF_1:def 1;
      then mm2 = -mm;
      hence thesis;
    end;
  end;
