reserve x, y, y1, y2 for object;
reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;
reserve u, v for VECTOR of V;
reserve i, j, k, n for Element of NAT;

theorem LMThFRat31:
  for s being Element of Rat-Module holds Lin{s} <> Rat-Module
  proof
    set ZS = Rat-Module;
    let s be Element of ZS;
    assume AS: Lin{s} = ZS;
    consider m, n be Integer such that
    P2: n > 0 & s = m/n by RAT_1:1;
      per cases;
      suppose m = 1 or m = -1;
        then per cases;
        suppose D1: m = 1;
          reconsider t = 1 /(n+1) as Element of ZS by RAT_1:def 1;
          t in Lin{s} by AS;
          then consider l be Linear_Combination of {s} such that
P3:       t = Sum(l) by ZMODUL02:64;
P4:       Sum(l) = l.s * s by ZMODUL02:21;
          reconsider k = l.s as Integer;
P5:       1 /(n+1) = k/n by D1,P2,P3,P4,LMTFRat2;
          k = n/(n+1) by P2,P5,XCMPLX_1:87;
          hence contradiction by LMThFRat31X,P2;
        end;
        suppose D2: m = -1;
          reconsider t = (-1) /(n+1) as Element of ZS by RAT_1:def 1;
          t in Lin{s} by AS;
          then consider l be Linear_Combination of {s} such that
P3:       t = Sum(l) by ZMODUL02:64;
P4:       Sum(l) = l.s * s by ZMODUL02:21;
          reconsider k = l.s as Integer;
          (-1) /(n+1) = k* (m/n) by P2,P3,P4,LMTFRat2
          .= -k/n by D2; then
          k = n/(n+1) by P2,XCMPLX_1:87;
          hence contradiction by LMThFRat31X,P2;
        end;
      end;
      suppose C2: m <>1 & m <> -1;
        reconsider t = (m+1)/n as Element of ZS by RAT_1:def 1;
        t in Lin{s} by AS;
        then consider l be Linear_Combination of {s} such that
        P3: t = Sum(l) by ZMODUL02:64;
        P4: Sum(l) = l.s * s by ZMODUL02:21;
        reconsider k = l.s as Integer;
        P5: (m+1)/n = k* (m/n) by P2,P3,P4,LMTFRat2;
        m+1 = (m+1)/n * n by P2,XCMPLX_1:87
        .= k* ((m/n) *n) by P5
        .= k*m by P2,XCMPLX_1:87;
        then m*(k-1) = 1;
        hence contradiction by C2,INT_1:9;
    end;
  end;
