set X = { l where l is Linear_Combination of A : verum } ;
{ l where l is Linear_Combination of A : verum } c= the carrier of (LC_Z_Module V)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { l where l is Linear_Combination of A : verum } or x in the carrier of (LC_Z_Module V) )
assume x in { l where l is Linear_Combination of A : verum } ; :: thesis: x in the carrier of (LC_Z_Module V)
then ex l being Linear_Combination of A st x = l ;
hence x in the carrier of (LC_Z_Module V) by Def29; :: thesis: verum

end;

then reconsider X = { l where l is Linear_Combination of A : verum } as Subset of (LC_Z_Module V) ;
A1: X is linearly-closed

proof

thus for v, u being Vector of (LC_Z_Module V) st v in X & u in X holds
v + u in X :: according to VECTSP_4:def 1 :: thesis: for b₁ being Element of the carrier of R
for b₂ being Element of the carrier of (LC_Z_Module V) holds
( not b₂ in X or b₁ * b₂ in X )

proof

let v, u be Vector of (LC_Z_Module V); :: thesis: ( v in X & u in X implies v + u in X )
assume that
A2: v in X and
A3: u in X ; :: thesis: v + u in X
consider l1 being Linear_Combination of A such that
A4: v = l1 by A2;
consider l2 being Linear_Combination of A such that
A5: u = l2 by A3;
A6: u = vector ((LC_Z_Module V),l2) by A5, RLVECT_2:def 1, RLVECT_1:1;
v = vector ((LC_Z_Module V),l1) by A4, RLVECT_2:def 1, RLVECT_1:1;
then v + u = l1 + l2 by A6, Th47;
then v + u is Linear_Combination of A by Th27;
hence v + u in X ; :: thesis: verum

end;

let a be Element of R; :: thesis: for b₁ being Element of the carrier of (LC_Z_Module V) holds
( not b₁ in X or a * b₁ in X )
let v be Vector of (LC_Z_Module V); :: thesis: ( not v in X or a * v in X )
assume v in X ; :: thesis: a * v in X
then consider l being Linear_Combination of A such that
A7: v = l ;
a * v = a * (vector ((LC_Z_Module V),l)) by A7, RLVECT_2:def 1, RLVECT_1:1
.= a * l by Th48 ;
then a * v is Linear_Combination of A by Th31;
hence a * v in X ; :: thesis: verum

end;

ZeroLC V is Linear_Combination of A by Th11;
then ZeroLC V in X ;
hence ex b₁ being strict Submodule of LC_Z_Module V st the carrier of b₁ = { l where l is Linear_Combination of A : verum } by A1, ZMODUL01:50; :: thesis: verum