reserve i,j for Nat;
reserve A,B for Ring;
reserve K, L for Field;

theorem
  for x be Element of F_Complex holds F_Rat is Subring of FQ_Ring(x)
proof
  let x be Element of F_Complex;
A1: the addF of F_Rat = (the addF of FQ_Ring(x))||the carrier of F_Rat
    by Lm57;
A2: the multF of F_Rat = (the multF of FQ_Ring(x))||the carrier of F_Rat
    by Lm58;
A3: 1.FQ_Ring(x) = 1.F_Complex by Lm52 .= 1.F_Rat by C0SP1:def 3,Th3;
    0.FQ_Ring(x) = 0.F_Rat by Lm48,Lm7,SUBSET_1:def 8;
    hence thesis by Lm55,A1,A2,A3,C0SP1:def 3;
end;
