let F be Field; :: thesis: for V being VectSp of F
for W being Subspace of V
for L being Linear_Compl of W
for v being Element of V
for t being Element of [:the carrier of V,the carrier of V:] st (t `1 ) + (t `2 ) = v & t `1 in W & t `2 in L holds
t = v |-- W,L
let V be VectSp of F; :: thesis: for W being Subspace of V
for L being Linear_Compl of W
for v being Element of V
for t being Element of [:the carrier of V,the carrier of V:] st (t `1 ) + (t `2 ) = v & t `1 in W & t `2 in L holds
t = v |-- W,L
let W be Subspace of V; :: thesis: for L being Linear_Compl of W
for v being Element of V
for t being Element of [:the carrier of V,the carrier of V:] st (t `1 ) + (t `2 ) = v & t `1 in W & t `2 in L holds
t = v |-- W,L
let L be Linear_Compl of W; :: thesis: for v being Element of V
for t being Element of [:the carrier of V,the carrier of V:] st (t `1 ) + (t `2 ) = v & t `1 in W & t `2 in L holds
t = v |-- W,L
let v be Element of V; :: thesis: for t being Element of [:the carrier of V,the carrier of V:] st (t `1 ) + (t `2 ) = v & t `1 in W & t `2 in L holds
t = v |-- W,L
let t be Element of [:the carrier of V,the carrier of V:]; :: thesis: ( (t `1 ) + (t `2 ) = v & t `1 in W & t `2 in L implies t = v |-- W,L )
V is_the_direct_sum_of W,L by Th48;
hence ( (t `1 ) + (t `2 ) = v & t `1 in W & t `2 in L implies t = v |-- W,L ) by Def6; :: thesis: verum