let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: 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; :: thesis: 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; :: thesis: ( ((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; :: thesis: verum