set W = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #);
A1: ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is vector-distributive

proof

let x be Element of GF; :: according to VECTSP_1:def 13 :: thesis: for b₁, b₂ being Element of the carrier of ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) holds x * (b₁ + b₂) = (x * b₁) + (x * b₂)
let v, w be Element of ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #); :: thesis: x * (v + w) = (x * v) + (x * w)
reconsider v9 = v, w9 = w as Element of V ;
thus x * (v + w) = x * (v9 + w9)
.= (x * v9) + (x * w9) by VECTSP_1:def 14
.= (x * v) + (x * w) ; :: thesis: verum

end;

A2: ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is scalar-distributive

proof

let x, y be Element of GF; :: according to VECTSP_1:def 14 :: thesis: for b₁ being Element of the carrier of ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) holds (x + y) * b₁ = (x * b₁) + (y * b₁)
let v be Element of ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #); :: thesis: (x + y) * v = (x * v) + (y * v)
reconsider v9 = v as Element of V ;
thus (x + y) * v = (x + y) * v9
.= (x * v9) + (y * v9) by VECTSP_1:def 15
.= (x * v) + (y * v) ; :: thesis: verum

end;

A3: ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is scalar-associative

proof

let x, y be Element of GF; :: according to VECTSP_1:def 15 :: thesis: for b₁ being Element of the carrier of ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) holds (x * y) * b₁ = x * (y * b₁)
let v be Element of ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #); :: thesis: (x * y) * v = x * (y * v)
reconsider v9 = v as Element of V ;
thus (x * y) * v = (x * y) * v9
.= x * (y * v9) by VECTSP_1:def 16
.= x * (y * v) ; :: thesis: verum

end;

A4: ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is scalar-unital

proof

let v be Element of ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #); :: according to VECTSP_1:def 16 :: thesis: (1. GF) * v = v
reconsider v9 = v as Element of V ;
thus (1. GF) * v = (1_ GF) * v9
.= v ; :: thesis: verum

end;

A5: for a being Element of GF
for v, w being Element of ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #)
for v9, w9 being Element of V st v = v9 & w = w9 holds
( v + w = v9 + w9 & a * v = a * v9 ) ;
( ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is Abelian & ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is add-associative & ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is right_zeroed & ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is right_complementable )

proof

thus ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is Abelian :: thesis: ( ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is add-associative & ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is right_zeroed & ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is right_complementable )

proof

let x, y be Element of ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #); :: according to RLVECT_1:def 2 :: thesis: x + y = y + x
reconsider x9 = x, y9 = y as Element of V ;
thus x + y = y9 + x9 by A5
.= y + x ; :: thesis: verum

end;

hereby :: according to RLVECT_1:def 3 :: thesis: ( ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is right_zeroed & ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is right_complementable )

let x, y, z be Element of ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #); :: thesis: (x + y) + z = x + (y + z)
reconsider x9 = x, y9 = y, z9 = z as Element of V ;
thus (x + y) + z = (x9 + y9) + z9
.= x9 + (y9 + z9) by RLVECT_1:def 3
.= x + (y + z) ; :: thesis: verum

end;

hereby :: according to RLVECT_1:def 4 :: thesis: ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is right_complementable

let x be Element of ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #); :: thesis: x + (0. ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #)) = x
reconsider x9 = x as Element of V ;
thus x + (0. ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #)) = x9 + (0. V)
.= x by RLVECT_1:4 ; :: thesis: verum

end;

let x be Element of ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #); :: according to ALGSTR_0:def 16 :: thesis: x is right_complementable
reconsider x9 = x as Element of V ;
consider b being Element of V such that
A6: x9 + b = 0. V by ALGSTR_0:def 11;
reconsider b9 = b as Element of ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) ;
take b9 ; :: according to ALGSTR_0:def 11 :: thesis: x + b9 = 0. ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #)
thus x + b9 = 0. ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) by A6; :: thesis: verum

end;

then reconsider W = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) as non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF by A4, A1, A2, A3;
A7: the lmult of W = the lmult of V | [: the carrier of GF, the carrier of W:] by RELSET_1:19;
( 0. W = 0. V & the addF of W = the addF of V || the carrier of W ) by RELSET_1:19;
hence ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) is strict Subspace of V by A7, Def2; :: thesis: verum