let K be Ring; :: thesis: for V being LeftMod of K holds

( SubJoin V is commutative & SubJoin V is associative & SubJoin V is having_a_unity & (0). V = the_unity_wrt (SubJoin V) )

let V be LeftMod of K; :: thesis: ( SubJoin V is commutative & SubJoin V is associative & SubJoin V is having_a_unity & (0). V = the_unity_wrt (SubJoin V) )

set S0 = Submodules V;

set S1 = SubJoin V;

reconsider L = LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) as Lattice by VECTSP_5:57;

SubJoin V = the L_join of L ;

hence ( SubJoin V is commutative & SubJoin V is associative ) ; :: thesis: ( SubJoin V is having_a_unity & (0). V = the_unity_wrt (SubJoin V) )

set e = (0). V;

reconsider e9 = @ ((0). V) as Element of Submodules V ;

A1: e9 = (0). V by LMOD_6:def 2;

hence SubJoin V is having_a_unity by SETWISEO:def 2; :: thesis: (0). V = the_unity_wrt (SubJoin V)

thus (0). V = the_unity_wrt (SubJoin V) by A1, A2, BINOP_1:def 8; :: thesis: verum

( SubJoin V is commutative & SubJoin V is associative & SubJoin V is having_a_unity & (0). V = the_unity_wrt (SubJoin V) )

let V be LeftMod of K; :: thesis: ( SubJoin V is commutative & SubJoin V is associative & SubJoin V is having_a_unity & (0). V = the_unity_wrt (SubJoin V) )

set S0 = Submodules V;

set S1 = SubJoin V;

reconsider L = LattStr(# (Submodules V),(SubJoin V),(SubMeet V) #) as Lattice by VECTSP_5:57;

SubJoin V = the L_join of L ;

hence ( SubJoin V is commutative & SubJoin V is associative ) ; :: thesis: ( SubJoin V is having_a_unity & (0). V = the_unity_wrt (SubJoin V) )

set e = (0). V;

reconsider e9 = @ ((0). V) as Element of Submodules V ;

A1: e9 = (0). V by LMOD_6:def 2;

now :: thesis: for a9 being Element of Submodules V holds

( (SubJoin V) . (e9,a9) = a9 & (SubJoin V) . (a9,e9) = a9 )

then A2:
e9 is_a_unity_wrt SubJoin V
by BINOP_1:3;( (SubJoin V) . (e9,a9) = a9 & (SubJoin V) . (a9,e9) = a9 )

let a9 be Element of Submodules V; :: thesis: ( (SubJoin V) . (e9,a9) = a9 & (SubJoin V) . (a9,e9) = a9 )

reconsider b = a9 as Element of Submodules V ;

reconsider a = b as strict Subspace of V ;

thus (SubJoin V) . (e9,a9) = ((0). V) + a by A1, VECTSP_5:def 7

.= a9 by VECTSP_5:9 ; :: thesis: (SubJoin V) . (a9,e9) = a9

thus (SubJoin V) . (a9,e9) = a + ((0). V) by A1, VECTSP_5:def 7

.= a9 by VECTSP_5:9 ; :: thesis: verum

end;reconsider b = a9 as Element of Submodules V ;

reconsider a = b as strict Subspace of V ;

thus (SubJoin V) . (e9,a9) = ((0). V) + a by A1, VECTSP_5:def 7

.= a9 by VECTSP_5:9 ; :: thesis: (SubJoin V) . (a9,e9) = a9

thus (SubJoin V) . (a9,e9) = a + ((0). V) by A1, VECTSP_5:def 7

.= a9 by VECTSP_5:9 ; :: thesis: verum

hence SubJoin V is having_a_unity by SETWISEO:def 2; :: thesis: (0). V = the_unity_wrt (SubJoin V)

thus (0). V = the_unity_wrt (SubJoin V) by A1, A2, BINOP_1:def 8; :: thesis: verum