:: Operations on Subspaces in Real Unitary Space
:: by Noboru Endou , Takashi Mitsuishi and Yasunari Shidama
::
:: Received October 9, 2002
:: Copyright (c) 2002-2012 Association of Mizar Users


begin

definition
let V be RealUnitarySpace;
let W1, W2 be Subspace of V;
func W1 + W2 -> strict Subspace of V means :Def1: :: RUSUB_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 ) }
proof end;
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 RUSUB_1:24;
end;

:: deftheorem Def1 defines + RUSUB_2:def 1 :
for V being RealUnitarySpace
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 RealUnitarySpace;
let W1, W2 be Subspace of V;
func W1 /\ W2 -> strict Subspace of V means :Def2: :: RUSUB_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
proof end;
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 RUSUB_1:24;
end;

:: deftheorem Def2 defines /\ RUSUB_2:def 2 :
for V being RealUnitarySpace
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 );

begin

theorem Th1: :: RUSUB_2:1
for V being RealUnitarySpace
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 ) )
proof end;

theorem Th2: :: RUSUB_2:2
for V being RealUnitarySpace
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
proof end;

theorem Th3: :: RUSUB_2:3
for V being RealUnitarySpace
for W1, W2 being Subspace of V
for x being set holds
( x in W1 /\ W2 iff ( x in W1 & x in W2 ) )
proof end;

Lm1: for V being RealUnitarySpace
for W1, W2 being Subspace of V holds W1 + W2 = W2 + W1
proof end;

Lm2: for V being RealUnitarySpace
for W1, W2 being Subspace of V holds the carrier of W1 c= the carrier of (W1 + W2)
proof end;

Lm3: for V being RealUnitarySpace
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
proof end;

theorem :: RUSUB_2:4
for V being RealUnitarySpace
for W being strict Subspace of V holds W + W = W by Lm3;

theorem :: RUSUB_2:5
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds W1 + W2 = W2 + W1 by Lm1;

theorem Th6: :: RUSUB_2:6
for V being RealUnitarySpace
for W1, W2, W3 being Subspace of V holds W1 + (W2 + W3) = (W1 + W2) + W3
proof end;

theorem Th7: :: RUSUB_2:7
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds
( W1 is Subspace of W1 + W2 & W2 is Subspace of W1 + W2 )
proof end;

theorem Th8: :: RUSUB_2:8
for V being RealUnitarySpace
for W1 being Subspace of V
for W2 being strict Subspace of V holds
( W1 is Subspace of W2 iff W1 + W2 = W2 )
proof end;

theorem Th9: :: RUSUB_2:9
for V being RealUnitarySpace
for W being strict Subspace of V holds
( ((0). V) + W = W & W + ((0). V) = W )
proof end;

theorem Th10: :: RUSUB_2:10
for V being RealUnitarySpace holds
( ((0). V) + ((Omega). V) = UNITSTR(# the carrier of V, the ZeroF of V, the U7 of V, the Mult of V, the scalar of V #) & ((Omega). V) + ((0). V) = UNITSTR(# the carrier of V, the ZeroF of V, the U7 of V, the Mult of V, the scalar of V #) )
proof end;

theorem Th11: :: RUSUB_2:11
for V being RealUnitarySpace
for W being Subspace of V holds
( ((Omega). V) + W = UNITSTR(# the carrier of V, the ZeroF of V, the U7 of V, the Mult of V, the scalar of V #) & W + ((Omega). V) = UNITSTR(# the carrier of V, the ZeroF of V, the U7 of V, the Mult of V, the scalar of V #) )
proof end;

theorem :: RUSUB_2:12
for V being strict RealUnitarySpace holds ((Omega). V) + ((Omega). V) = V by Th11;

theorem :: RUSUB_2:13
for V being RealUnitarySpace
for W being strict Subspace of V holds W /\ W = W
proof end;

theorem Th14: :: RUSUB_2:14
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds W1 /\ W2 = W2 /\ W1
proof end;

theorem Th15: :: RUSUB_2:15
for V being RealUnitarySpace
for W1, W2, W3 being Subspace of V holds W1 /\ (W2 /\ W3) = (W1 /\ W2) /\ W3
proof end;

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

theorem Th16: :: RUSUB_2:16
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds
( W1 /\ W2 is Subspace of W1 & W1 /\ W2 is Subspace of W2 )
proof end;

theorem Th17: :: RUSUB_2:17
for V being RealUnitarySpace
for W2 being Subspace of V
for W1 being strict Subspace of V holds
( W1 is Subspace of W2 iff W1 /\ W2 = W1 )
proof end;

theorem Th18: :: RUSUB_2:18
for V being RealUnitarySpace
for W being Subspace of V holds
( ((0). V) /\ W = (0). V & W /\ ((0). V) = (0). V )
proof end;

theorem :: RUSUB_2:19
for V being RealUnitarySpace holds
( ((0). V) /\ ((Omega). V) = (0). V & ((Omega). V) /\ ((0). V) = (0). V ) by Th18;

theorem Th20: :: RUSUB_2:20
for V being RealUnitarySpace
for W being strict Subspace of V holds
( ((Omega). V) /\ W = W & W /\ ((Omega). V) = W )
proof end;

theorem :: RUSUB_2:21
for V being strict RealUnitarySpace holds ((Omega). V) /\ ((Omega). V) = V
proof end;

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

theorem :: RUSUB_2:22
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds W1 /\ W2 is Subspace of W1 + W2 by Lm5, RUSUB_1:22;

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

theorem :: RUSUB_2:23
for V being RealUnitarySpace
for W1 being Subspace of V
for W2 being strict Subspace of V holds (W1 /\ W2) + W2 = W2 by Lm6, RUSUB_1:24;

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

theorem :: RUSUB_2:24
for V being RealUnitarySpace
for W1 being Subspace of V
for W2 being strict Subspace of V holds W2 /\ (W2 + W1) = W2 by Lm7, RUSUB_1:24;

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

theorem :: RUSUB_2:25
for V being RealUnitarySpace
for W1, W2, W3 being Subspace of V holds (W1 /\ W2) + (W2 /\ W3) is Subspace of W2 /\ (W1 + W3) by Lm8, RUSUB_1:22;

Lm9: for V being RealUnitarySpace
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))
proof end;

theorem :: RUSUB_2:26
for V being RealUnitarySpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 holds
W2 /\ (W1 + W3) = (W1 /\ W2) + (W2 /\ W3) by Lm9, RUSUB_1:24;

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

theorem :: RUSUB_2:27
for V being RealUnitarySpace
for W1, W2, W3 being Subspace of V holds W2 + (W1 /\ W3) is Subspace of (W1 + W2) /\ (W2 + W3) by Lm10, RUSUB_1:22;

Lm11: for V being RealUnitarySpace
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))
proof end;

theorem :: RUSUB_2:28
for V being RealUnitarySpace
for W1, W2, W3 being Subspace of V st W1 is Subspace of W2 holds
W2 + (W1 /\ W3) = (W1 + W2) /\ (W2 + W3) by Lm11, RUSUB_1:24;

theorem Th29: :: RUSUB_2:29
for V being RealUnitarySpace
for W1, W2, W3 being Subspace of V st W1 is strict Subspace of W3 holds
W1 + (W2 /\ W3) = (W1 + W2) /\ W3
proof end;

theorem :: RUSUB_2:30
for V being RealUnitarySpace
for W1, W2 being strict Subspace of V holds
( W1 + W2 = W2 iff W1 /\ W2 = W1 )
proof end;

theorem :: RUSUB_2:31
for V being RealUnitarySpace
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
proof end;

theorem :: RUSUB_2:32
for V being RealUnitarySpace
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 )
proof end;

begin

definition
let V be RealUnitarySpace;
func Subspaces V -> set means :Def3: :: RUSUB_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 )
proof end;
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
proof end;
end;

:: deftheorem Def3 defines Subspaces RUSUB_2:def 3 :
for V being RealUnitarySpace
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 RealUnitarySpace;
cluster Subspaces V -> non empty ;
coherence
not Subspaces V is empty
proof end;
end;

theorem :: RUSUB_2:33
for V being strict RealUnitarySpace holds V in Subspaces V
proof end;

begin

definition
let V be RealUnitarySpace;
let W1, W2 be Subspace of V;
pred V is_the_direct_sum_of W1,W2 means :Def4: :: RUSUB_2:def 4
( UNITSTR(# the carrier of V, the ZeroF of V, the U7 of V, the Mult of V, the scalar of V #) = W1 + W2 & W1 /\ W2 = (0). V );
end;

:: deftheorem Def4 defines is_the_direct_sum_of RUSUB_2:def 4 :
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds
( V is_the_direct_sum_of W1,W2 iff ( UNITSTR(# the carrier of V, the ZeroF of V, the U7 of V, the Mult of V, the scalar of V #) = W1 + W2 & W1 /\ W2 = (0). V ) );

Lm12: for V being RealUnitarySpace
for W being strict Subspace of V st ( for v being VECTOR of V holds v in W ) holds
W = UNITSTR(# the carrier of V, the ZeroF of V, the U7 of V, the Mult of V, the scalar of V #)
proof end;

Lm13: for V being RealUnitarySpace
for W1, W2 being Subspace of V holds
( W1 + W2 = UNITSTR(# the carrier of V, the ZeroF of V, the U7 of V, the Mult of V, the scalar 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 ) )
proof end;

Lm14: for V being RealUnitarySpace
for W being Subspace of V ex C being strict Subspace of V st V is_the_direct_sum_of C,W
proof end;

definition
let V be RealUnitarySpace;
let W be Subspace of V;
mode Linear_Compl of W -> Subspace of V means :Def5: :: RUSUB_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
proof end;
end;

:: deftheorem Def5 defines Linear_Compl RUSUB_2:def 5 :
for V being RealUnitarySpace
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 RealUnitarySpace;
let W be Subspace of V;
cluster non empty V99() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital strict RealUnitarySpace-like for Linear_Compl of W;
existence
ex b1 being Linear_Compl of W st b1 is strict
proof end;
end;

Lm15: for V being RealUnitarySpace
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
proof end;

theorem :: RUSUB_2:34
for V being RealUnitarySpace
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
W2 is Linear_Compl of W1
proof end;

theorem Th35: :: RUSUB_2:35
for V being RealUnitarySpace
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 )
proof end;

begin

:: Theorems concerning the direct sum, linear complement
:: and coset of a subspace
theorem Th36: :: RUSUB_2:36
for V being RealUnitarySpace
for W being Subspace of V
for L being Linear_Compl of W holds
( W + L = UNITSTR(# the carrier of V, the ZeroF of V, the U7 of V, the Mult of V, the scalar of V #) & L + W = UNITSTR(# the carrier of V, the ZeroF of V, the U7 of V, the Mult of V, the scalar of V #) )
proof end;

theorem Th37: :: RUSUB_2:37
for V being RealUnitarySpace
for W being Subspace of V
for L being Linear_Compl of W holds
( W /\ L = (0). V & L /\ W = (0). V )
proof end;

theorem :: RUSUB_2:38
for V being RealUnitarySpace
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 Lm15;

theorem Th39: :: RUSUB_2:39
for V being RealUnitarySpace holds
( V is_the_direct_sum_of (0). V, (Omega). V & V is_the_direct_sum_of (Omega). V, (0). V )
proof end;

theorem :: RUSUB_2:40
for V being RealUnitarySpace
for W being Subspace of V
for L being Linear_Compl of W holds W is Linear_Compl of L
proof end;

theorem :: RUSUB_2:41
for V being RealUnitarySpace holds
( (0). V is Linear_Compl of (Omega). V & (Omega). V is Linear_Compl of (0). V )
proof end;

Lm16: for V being RealUnitarySpace
for W being Subspace of V
for v being VECTOR of V
for x being set holds
( x in v + W iff ex u being VECTOR of V st
( u in W & x = v + u ) )
proof end;

theorem Th42: :: RUSUB_2:42
for V being RealUnitarySpace
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
proof end;

Lm17: for V being RealUnitarySpace
for W being Subspace of V
for v being VECTOR of V ex C being Coset of W st v in C
proof end;

theorem Th43: :: RUSUB_2:43
for V being RealUnitarySpace
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} )
proof end;

begin

theorem :: RUSUB_2:44
for V being RealUnitarySpace
for W1, W2 being Subspace of V holds
( W1 + W2 = UNITSTR(# the carrier of V, the ZeroF of V, the U7 of V, the Mult of V, the scalar 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: :: RUSUB_2:45
for V being RealUnitarySpace
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 )
proof end;

theorem :: RUSUB_2:46
for V being RealUnitarySpace
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
proof end;

definition
let V be RealUnitarySpace;
let v be VECTOR of V;
let W1, W2 be Subspace of V;
assume A1: V is_the_direct_sum_of W1,W2 ;
func v |-- (W1,W2) -> Element of [: the carrier of V, the carrier of V:] means :Def6: :: RUSUB_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 )
proof end;
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
proof end;
end;

:: deftheorem Def6 defines |-- RUSUB_2:def 6 :
for V being RealUnitarySpace
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: :: RUSUB_2:47
for V being RealUnitarySpace
for v being VECTOR of V
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
(v |-- (W1,W2)) `1 = (v |-- (W2,W1)) `2
proof end;

theorem Th48: :: RUSUB_2:48
for V being RealUnitarySpace
for v being VECTOR of V
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
(v |-- (W1,W2)) `2 = (v |-- (W2,W1)) `1
proof end;

theorem :: RUSUB_2:49
for V being RealUnitarySpace
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)
proof end;

theorem :: RUSUB_2:50
for V being RealUnitarySpace
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
proof end;

theorem :: RUSUB_2:51
for V being RealUnitarySpace
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 )
proof end;

theorem :: RUSUB_2:52
for V being RealUnitarySpace
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
proof end;

theorem :: RUSUB_2:53
for V being RealUnitarySpace
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
proof end;

begin

definition
let V be RealUnitarySpace;
func SubJoin V -> BinOp of (Subspaces V) means :Def7: :: RUSUB_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
proof end;
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
proof end;
end;

:: deftheorem Def7 defines SubJoin RUSUB_2:def 7 :
for V being RealUnitarySpace
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 RealUnitarySpace;
func SubMeet V -> BinOp of (Subspaces V) means :Def8: :: RUSUB_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
proof end;
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
proof end;
end;

:: deftheorem Def8 defines SubMeet RUSUB_2:def 8 :
for V being RealUnitarySpace
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 );

begin

theorem Th54: :: RUSUB_2:54
for V being RealUnitarySpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is Lattice
proof end;

registration
let V be RealUnitarySpace;
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: :: RUSUB_2:55
for V being RealUnitarySpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is lower-bounded
proof end;

theorem Th56: :: RUSUB_2:56
for V being RealUnitarySpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is upper-bounded
proof end;

theorem Th57: :: RUSUB_2:57
for V being RealUnitarySpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is 01_Lattice
proof end;

theorem Th58: :: RUSUB_2:58
for V being RealUnitarySpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is modular
proof end;

theorem Th59: :: RUSUB_2:59
for V being RealUnitarySpace holds LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is complemented
proof end;

registration
let V be RealUnitarySpace;
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;

theorem :: RUSUB_2:60
for V being RealUnitarySpace
for W1, W2, W3 being strict Subspace of V st W1 is Subspace of W2 holds
W1 /\ W3 is Subspace of W2 /\ W3
proof end;

begin

theorem :: RUSUB_2:61
for V being RealUnitarySpace
for W being strict Subspace of V st ( for v being VECTOR of V holds v in W ) holds
W = UNITSTR(# the carrier of V, the ZeroF of V, the U7 of V, the Mult of V, the scalar of V #) by Lm12;

theorem :: RUSUB_2:62
for V being RealUnitarySpace
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 :: RUSUB_2:63
for V being RealUnitarySpace
for W being Subspace of V
for v being VECTOR of V
for x being set holds
( x in v + W iff ex u being VECTOR of V st
( u in W & x = v + u ) ) by Lm16;