 reserve V for Z_Module;
 reserve W for Subspace of V;
 reserve v, u for Vector of V;
 reserve i for Element of INT.Ring;

theorem LmISRank51X:
  for V being torsion-free Z_Module,
  W1, W2 be finite-rank free Subspace of V holds
  rank(W1 + W2) + rank(W1 /\ W2) = rank(W1) + rank(W2)
  proof
    let V be torsion-free Z_Module,
    W1, W2 be finite-rank free Subspace of V;
    set W1W2 = W1 + W2;
    reconsider W10 = W1 as finite-rank free Subspace of W1W2
    by ZMODUL01:97;
    reconsider W20 = W2 as finite-rank free Subspace of W1W2
    by ZMODUL01:97;
    R0: rank(W10+W20) + rank(W10 /\ W20) = rank(W10) + rank(W20)
    by LmISRank501;
    SB1: W10 + W20 is strict Subspace of V by ZMODUL01:42;
    for v be Vector of V holds v in W10+W20 iff v in W1+W2
    proof
      let v be Vector of V;
      hereby
        assume v in (W10 + W20);
        then consider v1, v2 be Vector of W1W2 such that
        S11: v1 in W10 & v2 in W20 & v = v1 + v2 by ZMODUL01:92;
        thus v in W1 + W2 by S11;
      end;
      assume v in W1 + W2;
      then consider v1, v2 be Vector of V such that
      S11: v1 in W1 & v2 in W2 & v = v1 + v2 by ZMODUL01:92;
      v1 in the carrier of W10 & v2 in the carrier of W20 by S11;
      then reconsider v11 = v1, v22 = v2 as Vector of W1W2 by ZMODUL01:25;
      v11+v22 = v1 + v2 by ZMODUL01:28;
      hence v in (W10+W20) by S11,ZMODUL01:92;
    end;
    then
    R1: rank(W10 + W20) = rank(W1 + W2) by SB1,ZMODUL01:46;
    SB2: W10/\ W20 is strict Subspace of V by ZMODUL01:42;
    for v be Vector of V holds v in W10 /\ W20 iff v in W1 /\ W2
    proof
      let v be Vector of V;
      hereby
        assume X1: v in  (W10 /\ W20);
        then reconsider v0 = v as Vector of W10 /\ W20;
        v0 in W10 & v0 in W20 by X1,ZMODUL01:94;
        hence v in (W1/\ W2) by ZMODUL01:94;
      end;
      assume X1:v in (W1 /\ W2);
      then reconsider v0 = v as Vector of W1 /\ W2;
      v in W1 & v in W2 by X1,ZMODUL01:94;
      hence v in (W10 /\ W20) by ZMODUL01:94;
    end;
    hence thesis by R1,R0,SB2,ZMODUL01:46;
  end;
