
theorem Th25:
  for K1 be Subfield of F_Rat holds
  the carrier of K1 = the carrier of F_Rat
  proof
    let K1 be Subfield of F_Rat;
    thus the carrier of K1 c= the carrier of F_Rat by EC_PF_1:def 1;
    let x be object;
    set C1 = the carrier of K1;
A1: INT c= C1 by Th24;
A2: NAT c= C1 by Th23;
    assume x in C;
    then reconsider x1 = x as Rational;
    consider m be Integer, k be Nat such that
    A3: k>0 & x1 = m/k by RAT_1:8;
    A4: m in C1 by A1,INT_1:def 2;
    reconsider k2 = k, m2 = m as Element of F_Rat by RAT_1:def 2;
    reconsider k0 = k as Element of K1 by A2,ORDINAL1:def 12;
    A5: k0 <> 0.K by A3;
    then k0 <> 0.K1 by EC_PF_1:def 1;
    then consider k0i be Element of K1 such that
    A6: k0i*k0 = 1.K1 by VECTSP_1:def 9;
    C1 c= C by EC_PF_1:def 1;
    then reconsider kk0 = k0i as Element of K;
    kk0*k2 = 1.K1 by A6,Th18
    .= 1.K by EC_PF_1:def 1;
    then A7: kk0 = k2" by A5,VECTSP_1:def 10;
    A8:the multF of K1 = (the multF of K) || C1 by EC_PF_1:def 1;
    m/k = m2/k2 by Th16,A3
    .= (the multF of K1).(m2,k2") by A4,A7,A8,FUNCT_1:49,ZFMISC_1:87;
    hence thesis by A3,A4,A7,BINOP_1:17;
  end;
