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 LMThFRat32:
  for s, t being Element of Rat-Module st s <> t
  holds not {s,t} is linearly-independent
  proof
    set ZS = Rat-Module;
    let s, t be Element of ZS;
    assume AS: s <> t;
    assume P1: {s,t} is linearly-independent;
    consider m, n be Integer such that
    P2: n > 0 & s = m/n by RAT_1:1;
    consider p, q be Integer such that
    P3: q > 0 & t = p/q by RAT_1:1;
    reconsider b = n*p as Element of INT.Ring by INT_1:def 2;
    reconsider a = q*m as Element of INT.Ring by INT_1:def 2;
    P4: p <> 0.INT.Ring by AS,P1,P3,ZMODUL02:62;
    b * s = (n*p)*(m/n) by LMTFRat2,P2
    .= p*m by P2,XCMPLX_1:90
    .= (q*m)*(p/q) by P3,XCMPLX_1:90
    .= a*t by LMTFRat2,P3;
    hence contradiction by AS,P1,P2,P4,ZMODUL02:62;
  end;
