:: Operations on Subspaces in Real Linear Space
:: by Wojciech A. Trybulec
::
:: Received September 20, 1989
:: Copyright (c) 1990-2012 Association of Mizar Users


begin

::
:: Definitions of sum and intersection of subspaces.
::
definition
let V be RealLinearSpace;
let W1, W2 be Subspace of V;
funcW1 + W2 ->  strict Subspace of V means :Def1: :: RLSUB_2:def 1
 the carrier of it =  {  (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 )  }  ;
existence
 ex b1 being strict Subspace of V st the carrier of b1 =  {  (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 )  }
proofend;
uniqueness
 for b1, b2 being strict Subspace of V st the carrier of b1 =  {  (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 )  }  & the carrier of b2 =  {  (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 )  }  holds
b1 = b2 by RLSUB_1:30;
end;

:: deftheorem Def1 defines + RLSUB_2:def_1_:_
 for V being RealLinearSpace
for W1, W2 being Subspace of V
for b4 being strict Subspace of V holds
( b4 = W1 + W2 iff the carrier of b4 =  {  (v + u) where v, u is VECTOR of V : ( v in W1 & u in W2 )  }  );

definition
let V be RealLinearSpace;
let W1, W2 be Subspace of V;
funcW1 /\ W2 ->  strict Subspace of V means :Def2: :: RLSUB_2:def 2
 the carrier of it = the carrier of W1 /\ the carrier of W2;
existence
 ex b1 being strict Subspace of V st the carrier of b1 = the carrier of W1 /\ the carrier of W2
proofend;
uniqueness
 for b1, b2 being strict Subspace of V st the carrier of b1 = the carrier of W1 /\ the carrier of W2 & the carrier of b2 = the carrier of W1 /\ the carrier of W2 holds
b1 = b2 by RLSUB_1:30;
end;

:: deftheorem Def2 defines /\ RLSUB_2:def_2_:_
 for V being RealLinearSpace
for W1, W2 being Subspace of V
for b4 being strict Subspace of V holds
( b4 = W1 /\ W2 iff the carrier of b4 = the carrier of W1 /\ the carrier of W2 );

::
:: Definitional theorems of sum and intersection of subspaces.
::
theorem Th1: :: RLSUB_2:1
for V being RealLinearSpace
for W1, W2 being Subspace of V
for x being set holds
( x in W1 + W2 iff ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & x = v1 + v2 ) )
proofend;

theorem Th2: :: RLSUB_2:2
for V being RealLinearSpace
for W1, W2 being Subspace of V
for v being VECTOR of V st ( v in W1 or v in W2 ) holds
v in W1 + W2
proofend;

theorem Th3: :: RLSUB_2:3
for V being RealLinearSpace
for W1, W2 being Subspace of V
for x being set holds
( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )
proofend;

Lm1:  for V being RealLinearSpace
for W1, W2 being Subspace of V holds W1 + W2 = W2 + W1
proofend;

Lm2:  for V being RealLinearSpace
for W1, W2 being Subspace of V holds the carrier of W1 c= the carrier of (W1 + W2)
proofend;

Lm3:  for V being RealLinearSpace
for W1 being Subspace of V
for W2 being strict Subspace of V st the carrier of W1 c= the carrier of W2 holds
W1 + W2 = W2
proofend;

theorem :: RLSUB_2:4
for V being RealLinearSpace
for W being strict Subspace of V holds W + W = W by Lm3;

theorem :: RLSUB_2:5
for V being RealLinearSpace
for W1, W2 being Subspace of V holds W1 + W2 = W2 + W1 by Lm1;

theorem Th6: :: RLSUB_2:6
for V being RealLinearSpace
for W1, W2, W3 being Subspace of V holds W1 + (W2 + W3) = (W1 + W2) + W3
proofend;

theorem Th7: :: RLSUB_2:7
for V being RealLinearSpace
for W1, W2 being Subspace of V holds
( W1 is Subspace of W1 + W2 & W2 is Subspace of W1 + W2 )
proofend;

theorem Th8: :: RLSUB_2:8
for V being RealLinearSpace
for W1 being Subspace of V
for W2 being strict Subspace of V holds
( W1 is Subspace of W2 iff W1 + W2 = W2 )
proofend;

theorem Th9: :: RLSUB_2:9
for V being RealLinearSpace
for W being strict Subspace of V holds
( ((0). V) + W = W & W + ((0). V) = W )
proofend;

theorem :: RLSUB_2:10
for V being RealLinearSpace holds
( ((0). V) + ((Omega). V) = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) & ((Omega). V) + ((0). V) = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) ) by Th9;

theorem Th11: :: RLSUB_2:11
for V being RealLinearSpace
for W being Subspace of V holds
( ((Omega). V) + W = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) & W + ((Omega). V) = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) )
proofend;

theorem :: RLSUB_2:12
for V being strict RealLinearSpace holds ((Omega). V) + ((Omega). V) = V by Th11;

theorem :: RLSUB_2:13
for V being RealLinearSpace
for W being strict Subspace of V holds W /\ W = W
proofend;

theorem Th14: :: RLSUB_2:14
for V being RealLinearSpace
for W1, W2 being Subspace of V holds W1 /\ W2 = W2 /\ W1
proofend;

theorem Th15: :: RLSUB_2:15
for V being RealLinearSpace
for W1, W2, W3 being Subspace of V holds W1 /\ (W2 /\ W3) = (W1 /\ W2) /\ W3
proofend;

Lm4:  for V being RealLinearSpace
for W1, W2 being Subspace of V holds the carrier of (W1 /\ W2) c= the carrier of W1
proofend;

theorem Th16: :: RLSUB_2:16
for V being RealLinearSpace
for W1, W2 being Subspace of V holds
( W1 /\ W2 is Subspace of W1 & W1 /\ W2 is Subspace of W2 )
proofend;

theorem Th17: :: RLSUB_2:17
for V being RealLinearSpace
for W2 being Subspace of V
for W1 being strict Subspace of V holds
( W1 is Subspace of W2 iff W1 /\ W2 = W1 )
proofend;

theorem Th18: :: RLSUB_2:18
for V being RealLinearSpace
for W being Subspace of V holds
( ((0). V) /\ W = (0). V & W /\ ((0). V) = (0). V )
proofend;

theorem :: RLSUB_2:19
for V being RealLinearSpace holds
( ((0). V) /\ ((Omega). V) = (0). V & ((Omega). V) /\ ((0). V) = (0). V ) by Th18;

theorem Th20: :: RLSUB_2:20
for V being RealLinearSpace
for W being strict Subspace of V holds
( ((Omega). V) /\ W = W & W /\ ((Omega). V) = W )
proofend;

theorem :: RLSUB_2:21
for V being strict RealLinearSpace holds ((Omega). V) /\ ((Omega). V) = V by Th20;

Lm5:  for V being RealLinearSpace
for W1, W2 being Subspace of V holds the carrier of (W1 /\ W2) c= the carrier of (W1 + W2)
proofend;

theorem :: RLSUB_2:22
for V being RealLinearSpace
for W1, W2 being Subspace of V holds W1 /\ W2 is Subspace of W1 + W2 by Lm5, RLSUB_1:28;

Lm6:  for V being RealLinearSpace
for W1, W2 being Subspace of V holds the carrier of ((W1 /\ W2) + W2) = the carrier of W2
proofend;

theorem :: RLSUB_2:23
for V being RealLinearSpace
for W1 being Subspace of V
for W2 being strict Subspace of V holds (W1 /\ W2) + W2 = W2 by Lm6, RLSUB_1:30;

Lm7:  for V being RealLinearSpace
for W1, W2 being Subspace of V holds the carrier of (W1 /\ (W1 + W2)) = the carrier of W1
proofend;

theorem :: RLSUB_2:24
for V being RealLinearSpace
for W2 being Subspace of V
for W1 being strict Subspace of V holds W1 /\ (W1 + W2) = W1 by Lm7, RLSUB_1:30;

Lm8:  for V being RealLinearSpace
for W1, W2, W3 being Subspace of V holds the carrier of ((W1 /\ W2) + (W2 /\ W3)) c= the carrier of (W2 /\ (W1 + W3))
proofend;

theorem :: RLSUB_2:25
for V being RealLinearSpace
for W1, W2, W3 being Subspace of V holds (W1 /\ W2) + (W2 /\ W3) is Subspace of W2 /\ (W1 + W3) by Lm8, RLSUB_1:28;

Lm9:  for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 holds
the carrier of (W2 /\ (W1 + W3)) = the carrier of ((W1 /\ W2) + (W2 /\ W3))
proofend;

theorem :: RLSUB_2:26
for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 holds
W2 /\ (W1 + W3) = (W1 /\ W2) + (W2 /\ W3) by Lm9, RLSUB_1:30;

Lm10:  for V being RealLinearSpace
for W2, W1, W3 being Subspace of V holds the carrier of (W2 + (W1 /\ W3)) c= the carrier of ((W1 + W2) /\ (W2 + W3))
proofend;

theorem :: RLSUB_2:27
for V being RealLinearSpace
for W2, W1, W3 being Subspace of V holds W2 + (W1 /\ W3) is Subspace of (W1 + W2) /\ (W2 + W3) by Lm10, RLSUB_1:28;

Lm11:  for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 holds
the carrier of (W2 + (W1 /\ W3)) = the carrier of ((W1 + W2) /\ (W2 + W3))
proofend;

theorem :: RLSUB_2:28
for V being RealLinearSpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 holds
W2 + (W1 /\ W3) = (W1 + W2) /\ (W2 + W3) by Lm11, RLSUB_1:30;

theorem Th29: :: RLSUB_2:29
for V being RealLinearSpace
for W1, W3, W2 being Subspace of V st W1 is strict Subspace of W3 holds
W1 + (W2 /\ W3) = (W1 + W2) /\ W3
proofend;

theorem :: RLSUB_2:30
for V being RealLinearSpace
for W1, W2 being strict Subspace of V holds
( W1 + W2 = W2 iff W1 /\ W2 = W1 )
proofend;

theorem :: RLSUB_2:31
for V being RealLinearSpace
for W1 being Subspace of V
for W2, W3 being strict Subspace of V st W1 is Subspace of W2 holds
W1 + W3 is Subspace of W2 + W3
proofend;

theorem :: RLSUB_2:32
for V being RealLinearSpace
for W1, W2 being Subspace of V holds
( ( W1 is Subspace of W2 or W2 is Subspace of W1 ) iff ex W being Subspace of V st the carrier of W = the carrier of W1 \/ the carrier of W2 )
proofend;

::
:: Introduction of a set of subspaces of real linear space.
::
definition
let V be RealLinearSpace;
func Subspaces V ->  set  means :Def3: :: RLSUB_2:def 3
 for x being set holds
( x in it iff x is strict Subspace of V );
existence
 ex b1 being set st
for x being set holds
( x in b1 iff x is strict Subspace of V )
proofend;
uniqueness
 for b1, b2 being set st ( for x being set holds
( x in b1 iff x is strict Subspace of V ) ) & ( for x being set holds
( x in b2 iff x is strict Subspace of V ) ) holds
b1 = b2
proofend;
end;

:: deftheorem Def3 defines Subspaces RLSUB_2:def_3_:_
 for V being RealLinearSpace
for b2 being set holds
( b2 = Subspaces V iff for x being set holds
( x in b2 iff x is strict Subspace of V ) );

registration
let V be RealLinearSpace;
cluster Subspaces V ->  non empty ;
coherence
 not Subspaces V is empty
proofend;
end;

theorem :: RLSUB_2:33
for V being strict RealLinearSpace holds V in Subspaces V
proofend;

::
:: Introduction of the direct sum of subspaces and
:: linear complement of subspace.
::
definition
let V be RealLinearSpace;
let W1, W2 be Subspace of V;
predV is_the_direct_sum_of W1,W2 means :Def4: :: RLSUB_2:def 4
( RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) = W1 + W2 & W1 /\ W2 = (0). V );
end;

:: deftheorem Def4 defines is_the_direct_sum_of RLSUB_2:def_4_:_
 for V being RealLinearSpace
for W1, W2 being Subspace of V holds
( V is_the_direct_sum_of W1,W2 iff ( RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) = W1 + W2 & W1 /\ W2 = (0). V ) );

Lm12:  for V being RealLinearSpace
for W being strict Subspace of V st ( for v being VECTOR of V holds v in W ) holds
W = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
proofend;

Lm13:  for V being RealLinearSpace
for W1, W2 being Subspace of V holds
( W1 + W2 = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) iff for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) )
proofend;

Lm14:  for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, v1, v2 being Element of V holds
( v = v1 + v2 iff v1 = v - v2 )
proofend;

Lm15:  for V being RealLinearSpace
for W being Subspace of V ex C being strict Subspace of V st V is_the_direct_sum_of C,W
proofend;

definition
let V be RealLinearSpace;
let W be Subspace of V;
mode Linear_Compl of W ->  Subspace of V means :Def5: :: RLSUB_2:def 5
V is_the_direct_sum_of it,W;
existence
 ex b1 being Subspace of V st V is_the_direct_sum_of b1,W
proofend;
end;

:: deftheorem Def5 defines Linear_Compl RLSUB_2:def_5_:_
 for V being RealLinearSpace
for W, b3 being Subspace of V holds
( b3 is Linear_Compl of W iff V is_the_direct_sum_of b3,W );

registration
let V be RealLinearSpace;
let W be Subspace of V;
cluster non empty left_complementable right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V131() for Linear_Compl of W;
existence
 ex b1 being Linear_Compl of W st b1 is strict
proofend;
end;

Lm16:  for V being RealLinearSpace
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
V is_the_direct_sum_of W2,W1
proofend;

theorem :: RLSUB_2:34
for V being RealLinearSpace
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
W2 is Linear_Compl of W1
proofend;

theorem Th35: :: RLSUB_2:35
for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W holds
( V is_the_direct_sum_of L,W & V is_the_direct_sum_of W,L )
proofend;

::
:: Theorems concerning the direct sum of a subspaces,
:: linear complement of a subspace and coset of a subspace.
::
theorem Th36: :: RLSUB_2:36
for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W holds
( W + L = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) & L + W = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) )
proofend;

theorem Th37: :: RLSUB_2:37
for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W holds
( W /\ L = (0). V & L /\ W = (0). V )
proofend;

theorem :: RLSUB_2:38
for V being RealLinearSpace
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
V is_the_direct_sum_of W2,W1 by Lm16;

theorem Th39: :: RLSUB_2:39
for V being RealLinearSpace holds
( V is_the_direct_sum_of (0). V, (Omega). V & V is_the_direct_sum_of (Omega). V, (0). V )
proofend;

theorem :: RLSUB_2:40
for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W holds W is Linear_Compl of L
proofend;

theorem :: RLSUB_2:41
for V being RealLinearSpace holds
( (0). V is Linear_Compl of (Omega). V & (Omega). V is Linear_Compl of (0). V )
proofend;

theorem Th42: :: RLSUB_2:42
for V being RealLinearSpace
for W1, W2 being Subspace of V
for C1 being Coset of W1
for C2 being Coset of W2 st C1 meets C2 holds
C1 /\ C2 is Coset of W1 /\ W2
proofend;

Lm17:  for V being RealLinearSpace
for W being Subspace of V
for v being VECTOR of V ex C being Coset of W st v in C
proofend;

theorem Th43: :: RLSUB_2:43
for V being RealLinearSpace
for W1, W2 being Subspace of V holds
( V is_the_direct_sum_of W1,W2 iff for C1 being Coset of W1
for C2 being Coset of W2 ex v being VECTOR of V st C1 /\ C2 = {v} )
proofend;

::
:: Decomposition of a vector.
::
theorem :: RLSUB_2:44
for V being RealLinearSpace
for W1, W2 being Subspace of V holds
( W1 + W2 = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) iff for v being VECTOR of V ex v1, v2 being VECTOR of V st
( v1 in W1 & v2 in W2 & v = v1 + v2 ) ) by Lm13;

theorem Th45: :: RLSUB_2:45
for V being RealLinearSpace
for W1, W2 being Subspace of V
for v, v1, v2, u1, u2 being VECTOR of V st V is_the_direct_sum_of W1,W2 & v = v1 + v2 & v = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 )
proofend;

theorem :: RLSUB_2:46
for V being RealLinearSpace
for W1, W2 being Subspace of V st V = W1 + W2 & ex v being VECTOR of V st
for v1, v2, u1, u2 being VECTOR of V st v = v1 + v2 & v = u1 + u2 & v1 in W1 & u1 in W1 & v2 in W2 & u2 in W2 holds
( v1 = u1 & v2 = u2 ) holds
V is_the_direct_sum_of W1,W2
proofend;

definition
let V be RealLinearSpace;
let v be VECTOR of V;
let W1, W2 be Subspace of V;
assume A1: V is_the_direct_sum_of W1,W2 ;
funcv |-- (W1,W2) ->  Element of [: the carrier of V, the carrier of V:] means :Def6: :: RLSUB_2:def 6
( v = (it `1) + (it `2) & it `1 in W1 & it `2 in W2 );
existence
 ex b1 being Element of [: the carrier of V, the carrier of V:] st
( v = (b1 `1) + (b1 `2) & b1 `1 in W1 & b1 `2 in W2 )
proofend;
uniqueness
 for b1, b2 being Element of [: the carrier of V, the carrier of V:] st v = (b1 `1) + (b1 `2) & b1 `1 in W1 & b1 `2 in W2 & v = (b2 `1) + (b2 `2) & b2 `1 in W1 & b2 `2 in W2 holds
b1 = b2
proofend;
end;

:: deftheorem Def6 defines |-- RLSUB_2:def_6_:_
 for V being RealLinearSpace
for v being VECTOR of V
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
for b5 being Element of [: the carrier of V, the carrier of V:] holds
( b5 = v |-- (W1,W2) iff ( v = (b5 `1) + (b5 `2) & b5 `1 in W1 & b5 `2 in W2 ) );

theorem Th47: :: RLSUB_2:47
for V being RealLinearSpace
for W1, W2 being Subspace of V
for v being VECTOR of V st V is_the_direct_sum_of W1,W2 holds
(v |-- (W1,W2)) `1 = (v |-- (W2,W1)) `2
proofend;

theorem Th48: :: RLSUB_2:48
for V being RealLinearSpace
for W1, W2 being Subspace of V
for v being VECTOR of V st V is_the_direct_sum_of W1,W2 holds
(v |-- (W1,W2)) `2 = (v |-- (W2,W1)) `1
proofend;

theorem :: RLSUB_2:49
for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W
for v being VECTOR of V
for t being Element of [: the carrier of V, the carrier of V:] st (t `1) + (t `2) = v & t `1 in W & t `2 in L holds
t = v |-- (W,L)
proofend;

theorem :: RLSUB_2:50
for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W
for v being VECTOR of V holds ((v |-- (W,L)) `1) + ((v |-- (W,L)) `2) = v
proofend;

theorem :: RLSUB_2:51
for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W
for v being VECTOR of V holds
( (v |-- (W,L)) `1 in W & (v |-- (W,L)) `2 in L )
proofend;

theorem :: RLSUB_2:52
for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W
for v being VECTOR of V holds (v |-- (W,L)) `1 = (v |-- (L,W)) `2
proofend;

theorem :: RLSUB_2:53
for V being RealLinearSpace
for W being Subspace of V
for L being Linear_Compl of W
for v being VECTOR of V holds (v |-- (W,L)) `2 = (v |-- (L,W)) `1
proofend;

definition
let V be RealLinearSpace;
func SubJoin V ->  BinOp of (Subspaces V) means :Def7: :: RLSUB_2:def 7
 for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
it . (A1,A2) = W1 + W2;
existence
 ex b1 being BinOp of (Subspaces V) st
for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
b1 . (A1,A2) = W1 + W2
proofend;
uniqueness
 for b1, b2 being BinOp of (Subspaces V) st ( for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
b1 . (A1,A2) = W1 + W2 ) & ( for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
b2 . (A1,A2) = W1 + W2 ) holds
b1 = b2
proofend;
end;

:: deftheorem Def7 defines SubJoin RLSUB_2:def_7_:_
 for V being RealLinearSpace
for b2 being BinOp of (Subspaces V) holds
( b2 = SubJoin V iff for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
b2 . (A1,A2) = W1 + W2 );

definition
let V be RealLinearSpace;
func SubMeet V ->  BinOp of (Subspaces V) means :Def8: :: RLSUB_2:def 8
 for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
it . (A1,A2) = W1 /\ W2;
existence
 ex b1 being BinOp of (Subspaces V) st
for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
b1 . (A1,A2) = W1 /\ W2
proofend;
uniqueness
 for b1, b2 being BinOp of (Subspaces V) st ( for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
b1 . (A1,A2) = W1 /\ W2 ) & ( for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
b2 . (A1,A2) = W1 /\ W2 ) holds
b1 = b2
proofend;
end;

:: deftheorem Def8 defines SubMeet RLSUB_2:def_8_:_
 for V being RealLinearSpace
for b2 being BinOp of (Subspaces V) holds
( b2 = SubMeet V iff for A1, A2 being Element of Subspaces V
for W1, W2 being Subspace of V st A1 = W1 & A2 = W2 holds
b2 . (A1,A2) = W1 /\ W2 );

::
:: Definitional theorems of functions SubJoin, SubMeet.
::
theorem Th54: :: RLSUB_2:54
for V being RealLinearSpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is Lattice
proofend;

registration
let V be RealLinearSpace;
cluster LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) ->  Lattice-like ;
coherence
 LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is Lattice-like by Th54;
end;

theorem Th55: :: RLSUB_2:55
for V being RealLinearSpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is lower-bounded
proofend;

theorem Th56: :: RLSUB_2:56
for V being RealLinearSpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is upper-bounded
proofend;

theorem Th57: :: RLSUB_2:57
for V being RealLinearSpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is 01_Lattice
proofend;

theorem Th58: :: RLSUB_2:58
for V being RealLinearSpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is modular
proofend;

Lm18:  for l being Lattice
for a, b being Element of l holds
( a is_a_complement_of b iff ( a "\/" b = Top l & a "/\" b = Bottom l ) )
proofend;

Lm19:  for l being Lattice
for b being Element of l st ( for a being Element of l holds a "/\" b = b ) holds
b = Bottom l
proofend;

Lm20:  for l being Lattice
for b being Element of l st ( for a being Element of l holds a "\/" b = b ) holds
b = Top l
proofend;

theorem Th59: :: RLSUB_2:59
for V being RealLinearSpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is complemented
proofend;

registration
let V be RealLinearSpace;
cluster LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) ->  modular lower-bounded upper-bounded complemented ;
coherence
 ( LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is lower-bounded & LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is upper-bounded & LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is modular & LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is complemented ) by Th55, Th56, Th58, Th59;
end;

::
:: Theorems concerning operations on subspaces (continuation). Proven
:: on the basis that set of subspaces with operations is a lattice.
::
theorem :: RLSUB_2:60
for V being RealLinearSpace
for W1, W2, W3 being strict Subspace of V st W1 is Subspace of W2 holds
W1 /\ W3 is Subspace of W2 /\ W3
proofend;

::
:: Auxiliary theorems.
::
theorem :: RLSUB_2:61
for V being non empty right_complementable add-associative right_zeroed addLoopStr
for v, v1, v2 being Element of V holds
( v = v1 + v2 iff v1 = v - v2 ) by Lm14;

theorem :: RLSUB_2:62
for V being RealLinearSpace
for W being strict Subspace of V st ( for v being VECTOR of V holds v in W ) holds
W = RLSStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) by Lm12;

theorem :: RLSUB_2:63
for V being RealLinearSpace
for W being Subspace of V
for v being VECTOR of V ex C being Coset of W st v in C by Lm17;

theorem :: RLSUB_2:64
for l being Lattice
for b being Element of l st ( for a being Element of l holds a "/\" b = b ) holds
b = Bottom l by Lm19;

theorem :: RLSUB_2:65
for l being Lattice
for b being Element of l st ( for a being Element of l holds a "\/" b = b ) holds
b = Top l by Lm20;