
theorem LmDE311A:
  for X being finite Subset of RAT
  ex a being Element of INT st
  a <> 0 &
  for r being Element of RAT st r in X holds a*r in INT
  proof
    defpred P[Nat] means
    for X being finite Subset of RAT st card X = $1 holds
    ex a being Element of INT st
    a <> 0 &
    for r being Element of RAT st r in X holds a*r in INT;
    P1: P[0]
    proof
      let X be finite Subset of RAT;
      assume AS: card X = 0;
      reconsider a = 1 as Element of INT by NUMBERS:17;
      take a;
      thus a <> 0;
      thus for r being Element of RAT st r in X holds a*r in INT by AS;
    end;
    P2: for n being Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      assume AS1: P[n];
      let X be finite Subset of RAT;
      assume AS2: card X = n+1;
      then X <> {};
      then consider x be object such that
      A1: x in X by XBOOLE_0:def 1;
      B6: {x} is Subset of X by A1,SUBSET_1:41;
      set Y = X \{x};
      reconsider Y as finite Subset of RAT;
      D1: X = Y \/ {x} by B6,XBOOLE_1:45;
      card Y = card X - card {x} by B6,CARD_2:44
      .= n+1 - 1 by AS2,CARD_1:30
      .= n;
      then consider a0 be Element of INT such that
      A4: a0 <> 0 and
      A5: for r being Element of RAT st r in Y holds a0*r in INT by AS1;
      reconsider x as Element of RAT by A1;
      consider x0, ib0 be Integer such that
      A6: ib0 > 0 & x = x0 / ib0 by RAT_1:1;
      reconsider ia0 = a0 as Integer;
      set ia = ia0*ib0;
      reconsider a = ia as Element of INT by INT_1:def 2;
      for r being Element of RAT st r in X holds a*r in INT
      proof
        let r be Element of RAT;
        assume r in X;
        then per cases by D1,XBOOLE_0:def 3;
        suppose r in Y;
          then reconsider iar = ia0*r as Integer by A5,INT_1:def 2;
          a*r = ib0*iar;
          hence a*r in INT by INT_1:def 2;
        end;
        suppose A72: r in {x};
          a*r = ia0 * (ib0*r)
          .= ia0 *(ib0*(x0/ib0)) by A6,A72,TARSKI:def 1
          .= ia0*x0 by A6,XCMPLX_1:87;
          hence a*r in INT by INT_1:def 2;
        end;
      end;
      hence thesis by A4,A6;
    end;
    P3: for n being Nat holds P[n] from NAT_1: sch 2(P1,P2);
    let X be finite Subset of RAT;
    card X is Nat;
    hence ex a being Element of INT st a <> 0 &
    for r being Element of RAT st r in X holds a*r in INT by P3;
  end;
