let V be torsion-free Z_Module; :: thesis: for W1, W2, W3 being free finite-rank Subspace of V st rank (W1 + W2) = rank W2 & W3 is Subspace of W1 holds
rank (W3 + W2) = rank W2
let W1, W2, W3 be free finite-rank Subspace of V; :: thesis: ( rank (W1 + W2) = rank W2 & W3 is Subspace of W1 implies rank (W3 + W2) = rank W2 )
assume A1: ( rank (W1 + W2) = rank W2 & W3 is Subspace of W1 ) ; :: thesis: rank (W3 + W2) = rank W2
for v being Vector of V st v in W3 + W2 holds
v in W1 + W2
proof
let v be Vector of V; :: thesis: ( v in W3 + W2 implies v in W1 + W2 )
assume B1: v in W3 + W2 ; :: thesis: v in W1 + W2
consider v1, v2 being Vector of V such that
B2: ( v1 in W3 & v2 in W2 & v = v1 + v2 ) by B1, ZMODUL01:92;
v1 in W1 by A1, B2, ZMODUL01:23;
hence v in W1 + W2 by B2, ZMODUL01:92; :: thesis: verum
end;
then W3 + W2 is Subspace of W1 + W2 by ZMODUL01:44;
then A2: rank (W3 + W2) <= rank (W1 + W2) by ZMODUL05:2;
W2 is Subspace of W3 + W2 by ZMODUL01:97;
then rank W2 <= rank (W3 + W2) by ZMODUL05:2;
hence rank (W3 + W2) = rank W2 by A1, A2, XXREAL_0:1; :: thesis: verum