let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; for M being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for W being Subspace of M holds
( ((Omega). M) + W = ModuleStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) & W + ((Omega). M) = ModuleStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) )
let M be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; for W being Subspace of M holds
( ((Omega). M) + W = ModuleStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) & W + ((Omega). M) = ModuleStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) )
let W be Subspace of M; ( ((Omega). M) + W = ModuleStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) & W + ((Omega). M) = ModuleStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) )
consider W9 being strict Subspace of M such that
A1:
the carrier of W9 = the carrier of ((Omega). M)
;
A2:
the carrier of W c= the carrier of W9
by A1, VECTSP_4:def 2;
A3:
W9 is Subspace of (Omega). M
by Lm6;
W + ((Omega). M) =
W + W9
by A1, Lm5
.=
W9
by A2, Lm3
.=
ModuleStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #)
by A1, A3, VECTSP_4:31
;
hence
( ((Omega). M) + W = ModuleStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) & W + ((Omega). M) = ModuleStr(# the carrier of M, the addF of M, the ZeroF of M, the lmult of M #) )
by Lm1; verum