let V be Z_Module; :: thesis: for W1, W2 being free Subspace of V
for I1 being Basis of W1
for I2 being Basis of W2 st V is_the_direct_sum_of W1,W2 holds
I1 /\ I2 = {}
let W1, W2 be free Subspace of V; :: thesis: for I1 being Basis of W1
for I2 being Basis of W2 st V is_the_direct_sum_of W1,W2 holds
I1 /\ I2 = {}
let I1 be Basis of W1; :: thesis: for I2 being Basis of W2 st V is_the_direct_sum_of W1,W2 holds
I1 /\ I2 = {}
let I2 be Basis of W2; :: thesis: ( V is_the_direct_sum_of W1,W2 implies I1 /\ I2 = {} )
assume A1: V is_the_direct_sum_of W1,W2 ; :: thesis: I1 /\ I2 = {}
assume I1 /\ I2 <> {} ; :: thesis: contradiction
then consider v being object such that
A2: v in I1 /\ I2 by XBOOLE_0:7;
A3: v in I1 by A2, XBOOLE_0:def 4;
not 0. W1 in I1 by ZMODUL02:57, VECTSP_7:def 3;
then A4: v <> 0. V by A3, ZMODUL01:26;
A5: v in W1 by A3;
v in W2 by A2;
then W1: v in W1 /\ W2 by A5, VECTSP_5:3;
W1 /\ W2 = (0). V by A1, VECTSP_5:def 4;
then v in (0). V by W1;
hence contradiction by A4, ZMODUL02:66; :: thesis: verum