let V be finite-dimensional RealUnitarySpace; :: thesis: for W1, W2 being Subspace of V holds (dim (W1 + W2)) + (dim (W1 /\ W2)) = (dim W1) + (dim W2)

let W1, W2 be Subspace of V; :: thesis: (dim (W1 + W2)) + (dim (W1 /\ W2)) = (dim W1) + (dim W2)

reconsider V = V as RealUnitarySpace ;

reconsider W1 = W1, W2 = W2 as Subspace of V ;

consider I being finite Subset of (W1 /\ W2) such that

A1: I is Basis of W1 /\ W2 by Def1;

W1 /\ W2 is Subspace of W2 by RUSUB_2:16;

then consider I2 being Basis of W2 such that

A2: I c= I2 by A1, RUSUB_3:24;

W1 /\ W2 is Subspace of W1 by RUSUB_2:16;

then consider I1 being Basis of W1 such that

A3: I c= I1 by A1, RUSUB_3:24;

reconsider I2 = I2 as finite Subset of W2 by Th3;

reconsider I1 = I1 as finite Subset of W1 by Th3;

I c= I1 /\ I2 by A3, A2, XBOOLE_1:19;

then I = I1 /\ I2 by A4;

then A15: card A = ((card I1) + (card I2)) - (card I) by CARD_2:45;

for L being Linear_Combination of A st Sum L = 0. (W1 + W2) holds

Carrier L = {}

the carrier of (W1 + W2) c= the carrier of V by RUSUB_1:def 1;

then reconsider A9 = A as Subset of V by XBOOLE_1:1;

A56: card I2 = dim W2 by Def2;

then ( Lin A9 = Lin A & W1 + W2 is Subspace of Lin A9 ) by RUSUB_1:22, RUSUB_3:28;

then Lin A = W1 + W2 by RUSUB_1:20;

then A67: A is Basis of W1 + W2 by A55, RUSUB_3:def 2;

card I = dim (W1 /\ W2) by A1, Def2;

then (dim (W1 + W2)) + (dim (W1 /\ W2)) = (((card I1) + (card I2)) + (- (card I))) + (card I) by A15, A67, Def2

.= (dim W1) + (dim W2) by A56, Def2 ;

hence (dim (W1 + W2)) + (dim (W1 /\ W2)) = (dim W1) + (dim W2) ; :: thesis: verum

let W1, W2 be Subspace of V; :: thesis: (dim (W1 + W2)) + (dim (W1 /\ W2)) = (dim W1) + (dim W2)

reconsider V = V as RealUnitarySpace ;

reconsider W1 = W1, W2 = W2 as Subspace of V ;

consider I being finite Subset of (W1 /\ W2) such that

A1: I is Basis of W1 /\ W2 by Def1;

W1 /\ W2 is Subspace of W2 by RUSUB_2:16;

then consider I2 being Basis of W2 such that

A2: I c= I2 by A1, RUSUB_3:24;

W1 /\ W2 is Subspace of W1 by RUSUB_2:16;

then consider I1 being Basis of W1 such that

A3: I c= I1 by A1, RUSUB_3:24;

reconsider I2 = I2 as finite Subset of W2 by Th3;

reconsider I1 = I1 as finite Subset of W1 by Th3;

A4: now :: thesis: I1 /\ I2 c= I

set A = I1 \/ I2;
I1 is linearly-independent
by RUSUB_3:def 2;

then reconsider I19 = I1 as linearly-independent Subset of V by RUSUB_3:22;

the carrier of (W1 /\ W2) c= the carrier of V by RUSUB_1:def 1;

then reconsider I9 = I as Subset of V by XBOOLE_1:1;

assume not I1 /\ I2 c= I ; :: thesis: contradiction

then consider x being object such that

A5: x in I1 /\ I2 and

A6: not x in I ;

x in I1 by A5, XBOOLE_0:def 4;

then x in Lin I1 by RUSUB_3:2;

then x in UNITSTR(# the carrier of W1, the ZeroF of W1, the addF of W1, the Mult of W1, the scalar of W1 #) by RUSUB_3:def 2;

then A7: x in the carrier of W1 ;

A8: the carrier of W1 c= the carrier of V by RUSUB_1:def 1;

then reconsider x9 = x as VECTOR of V by A7;

x in {x} by TARSKI:def 1;

then A11: x9 in Ix by XBOOLE_0:def 3;

x in I2 by A5, XBOOLE_0:def 4;

then x in Lin I2 by RUSUB_3:2;

then x in UNITSTR(# the carrier of W2, the ZeroF of W2, the addF of W2, the Mult of W2, the scalar of W2 #) by RUSUB_3:def 2;

then x in the carrier of W2 ;

then x in the carrier of W1 /\ the carrier of W2 by A7, XBOOLE_0:def 4;

then x in the carrier of (W1 /\ W2) by RUSUB_2:def 2;

then x in UNITSTR(# the carrier of (W1 /\ W2), the ZeroF of (W1 /\ W2), the addF of (W1 /\ W2), the Mult of (W1 /\ W2), the scalar of (W1 /\ W2) #) ;

then A12: x in Lin I by A1, RUSUB_3:def 2;

Ix \ {x} = I \ {x} by XBOOLE_1:40

.= I by A6, ZFMISC_1:57 ;

then not x9 in Lin I9 by A10, A11, RLVECT_3:5, RUSUB_3:25;

hence contradiction by A12, RUSUB_3:28; :: thesis: verum

end;then reconsider I19 = I1 as linearly-independent Subset of V by RUSUB_3:22;

the carrier of (W1 /\ W2) c= the carrier of V by RUSUB_1:def 1;

then reconsider I9 = I as Subset of V by XBOOLE_1:1;

assume not I1 /\ I2 c= I ; :: thesis: contradiction

then consider x being object such that

A5: x in I1 /\ I2 and

A6: not x in I ;

x in I1 by A5, XBOOLE_0:def 4;

then x in Lin I1 by RUSUB_3:2;

then x in UNITSTR(# the carrier of W1, the ZeroF of W1, the addF of W1, the Mult of W1, the scalar of W1 #) by RUSUB_3:def 2;

then A7: x in the carrier of W1 ;

A8: the carrier of W1 c= the carrier of V by RUSUB_1:def 1;

then reconsider x9 = x as VECTOR of V by A7;

now :: thesis: for y being object st y in I \/ {x} holds

y in the carrier of V

then reconsider Ix = I \/ {x} as Subset of V by TARSKI:def 3;y in the carrier of V

let y be object ; :: thesis: ( y in I \/ {x} implies y in the carrier of V )

the carrier of (W1 /\ W2) c= the carrier of V by RUSUB_1:def 1;

then A9: I c= the carrier of V ;

assume y in I \/ {x} ; :: thesis: y in the carrier of V

then ( y in I or y in {x} ) by XBOOLE_0:def 3;

then ( y in the carrier of V or y = x ) by A9, TARSKI:def 1;

hence y in the carrier of V by A7, A8; :: thesis: verum

end;the carrier of (W1 /\ W2) c= the carrier of V by RUSUB_1:def 1;

then A9: I c= the carrier of V ;

assume y in I \/ {x} ; :: thesis: y in the carrier of V

then ( y in I or y in {x} ) by XBOOLE_0:def 3;

then ( y in the carrier of V or y = x ) by A9, TARSKI:def 1;

hence y in the carrier of V by A7, A8; :: thesis: verum

now :: thesis: for y being object st y in I \/ {x} holds

y in I19

then A10:
Ix c= I19
;y in I19

let y be object ; :: thesis: ( y in I \/ {x} implies y in I19 )

assume y in I \/ {x} ; :: thesis: y in I19

then ( y in I or y in {x} ) by XBOOLE_0:def 3;

then ( y in I1 or y = x ) by A3, TARSKI:def 1;

hence y in I19 by A5, XBOOLE_0:def 4; :: thesis: verum

end;assume y in I \/ {x} ; :: thesis: y in I19

then ( y in I or y in {x} ) by XBOOLE_0:def 3;

then ( y in I1 or y = x ) by A3, TARSKI:def 1;

hence y in I19 by A5, XBOOLE_0:def 4; :: thesis: verum

x in {x} by TARSKI:def 1;

then A11: x9 in Ix by XBOOLE_0:def 3;

x in I2 by A5, XBOOLE_0:def 4;

then x in Lin I2 by RUSUB_3:2;

then x in UNITSTR(# the carrier of W2, the ZeroF of W2, the addF of W2, the Mult of W2, the scalar of W2 #) by RUSUB_3:def 2;

then x in the carrier of W2 ;

then x in the carrier of W1 /\ the carrier of W2 by A7, XBOOLE_0:def 4;

then x in the carrier of (W1 /\ W2) by RUSUB_2:def 2;

then x in UNITSTR(# the carrier of (W1 /\ W2), the ZeroF of (W1 /\ W2), the addF of (W1 /\ W2), the Mult of (W1 /\ W2), the scalar of (W1 /\ W2) #) ;

then A12: x in Lin I by A1, RUSUB_3:def 2;

Ix \ {x} = I \ {x} by XBOOLE_1:40

.= I by A6, ZFMISC_1:57 ;

then not x9 in Lin I9 by A10, A11, RLVECT_3:5, RUSUB_3:25;

hence contradiction by A12, RUSUB_3:28; :: thesis: verum

now :: thesis: for v being object st v in I1 \/ I2 holds

v in the carrier of (W1 + W2)

then reconsider A = I1 \/ I2 as finite Subset of (W1 + W2) by TARSKI:def 3;v in the carrier of (W1 + W2)

let v be object ; :: thesis: ( v in I1 \/ I2 implies v in the carrier of (W1 + W2) )

A13: ( the carrier of W1 c= the carrier of V & the carrier of W2 c= the carrier of V ) by RUSUB_1:def 1;

assume v in I1 \/ I2 ; :: thesis: v in the carrier of (W1 + W2)

then A14: ( v in I1 or v in I2 ) by XBOOLE_0:def 3;

then ( v in the carrier of W1 or v in the carrier of W2 ) ;

then reconsider v9 = v as VECTOR of V by A13;

( v9 in W1 or v9 in W2 ) by A14;

then v9 in W1 + W2 by RUSUB_2:2;

hence v in the carrier of (W1 + W2) ; :: thesis: verum

end;A13: ( the carrier of W1 c= the carrier of V & the carrier of W2 c= the carrier of V ) by RUSUB_1:def 1;

assume v in I1 \/ I2 ; :: thesis: v in the carrier of (W1 + W2)

then A14: ( v in I1 or v in I2 ) by XBOOLE_0:def 3;

then ( v in the carrier of W1 or v in the carrier of W2 ) ;

then reconsider v9 = v as VECTOR of V by A13;

( v9 in W1 or v9 in W2 ) by A14;

then v9 in W1 + W2 by RUSUB_2:2;

hence v in the carrier of (W1 + W2) ; :: thesis: verum

I c= I1 /\ I2 by A3, A2, XBOOLE_1:19;

then I = I1 /\ I2 by A4;

then A15: card A = ((card I1) + (card I2)) - (card I) by CARD_2:45;

for L being Linear_Combination of A st Sum L = 0. (W1 + W2) holds

Carrier L = {}

proof

then A55:
A is linearly-independent
by RLVECT_3:def 1;
( W1 is Subspace of W1 + W2 & I1 is linearly-independent )
by RUSUB_2:7, RUSUB_3:def 2;

then reconsider I19 = I1 as linearly-independent Subset of (W1 + W2) by RUSUB_3:22;

reconsider W29 = W2 as Subspace of W1 + W2 by RUSUB_2:7;

reconsider W19 = W1 as Subspace of W1 + W2 by RUSUB_2:7;

let L be Linear_Combination of A; :: thesis: ( Sum L = 0. (W1 + W2) implies Carrier L = {} )

assume A16: Sum L = 0. (W1 + W2) ; :: thesis: Carrier L = {}

A17: I1 misses (Carrier L) \ I1 by XBOOLE_1:79;

set B = (Carrier L) /\ I1;

consider F being FinSequence of the carrier of (W1 + W2) such that

A18: F is one-to-one and

A19: rng F = Carrier L and

A20: Sum L = Sum (L (#) F) by RLVECT_2:def 8;

reconsider B = (Carrier L) /\ I1 as Subset of (rng F) by A19, XBOOLE_1:17;

reconsider F1 = F - (B `), F2 = F - B as FinSequence of the carrier of (W1 + W2) by FINSEQ_3:86;

consider L1 being Linear_Combination of W1 + W2 such that

A21: Carrier L1 = (rng F1) /\ (Carrier L) and

A22: L1 (#) F1 = L (#) F1 by RLVECT_5:7;

F1 is one-to-one by A18, FINSEQ_3:87;

then A23: Sum (L (#) F1) = Sum L1 by A21, A22, RLVECT_5:6, XBOOLE_1:17;

rng F c= rng F ;

then reconsider X = rng F as Subset of (rng F) ;

consider L2 being Linear_Combination of W1 + W2 such that

A24: Carrier L2 = (rng F2) /\ (Carrier L) and

A25: L2 (#) F2 = L (#) F2 by RLVECT_5:7;

F2 is one-to-one by A18, FINSEQ_3:87;

then A26: Sum (L (#) F2) = Sum L2 by A24, A25, RLVECT_5:6, XBOOLE_1:17;

X \ (B `) = X /\ ((B `) `) by SUBSET_1:13

.= B by XBOOLE_1:28 ;

then rng F1 = B by FINSEQ_3:65;

then A27: Carrier L1 = I1 /\ ((Carrier L) /\ (Carrier L)) by A21, XBOOLE_1:16

.= (Carrier L) /\ I1 ;

then consider K1 being Linear_Combination of W19 such that

Carrier K1 = Carrier L1 and

A28: Sum K1 = Sum L1 by RUSUB_3:20;

rng F2 = (Carrier L) \ ((Carrier L) /\ I1) by A19, FINSEQ_3:65

.= (Carrier L) \ I1 by XBOOLE_1:47 ;

then A29: Carrier L2 = (Carrier L) \ I1 by A24, XBOOLE_1:28, XBOOLE_1:36;

then (Carrier L1) /\ (Carrier L2) = (Carrier L) /\ (I1 /\ ((Carrier L) \ I1)) by A27, XBOOLE_1:16

.= (Carrier L) /\ {} by A17

.= {} ;

then A30: Carrier L1 misses Carrier L2 ;

A31: Carrier L c= I1 \/ I2 by RLVECT_2:def 6;

then A32: Carrier L2 c= I2 by A29, XBOOLE_1:43;

Carrier L2 c= I2 by A31, A29, XBOOLE_1:43;

then consider K2 being Linear_Combination of W29 such that

Carrier K2 = Carrier L2 and

A33: Sum K2 = Sum L2 by RUSUB_3:20, XBOOLE_1:1;

A34: Sum K1 in W1 ;

ex P being Permutation of (dom F) st (F - (B `)) ^ (F - B) = F * P by FINSEQ_3:115;

then A35: 0. (W1 + W2) = Sum (L (#) (F1 ^ F2)) by A16, A20, RLVECT_5:4

.= Sum ((L (#) F1) ^ (L (#) F2)) by RLVECT_3:34

.= (Sum L1) + (Sum L2) by A23, A26, RLVECT_1:41 ;

then Sum L1 = - (Sum L2) by RLVECT_1:def 10

.= - (Sum K2) by A33, RUSUB_1:9 ;

then Sum K1 in W2 by A28;

then Sum K1 in W1 /\ W2 by A34, RUSUB_2:3;

then Sum K1 in Lin I by A1, RUSUB_3:def 2;

then consider KI being Linear_Combination of I such that

A36: Sum K1 = Sum KI by RUSUB_3:1;

A37: Carrier L = (Carrier L1) \/ (Carrier L2) by A27, A29, XBOOLE_1:51;

A47: I2 is linearly-independent by RUSUB_3:def 2;

A48: Carrier L1 c= I1 by A27, XBOOLE_1:17;

W1 /\ W2 is Subspace of W1 + W2 by RUSUB_2:22;

then consider LI being Linear_Combination of W1 + W2 such that

A49: Carrier LI = Carrier KI and

A50: Sum LI = Sum KI by RUSUB_3:19;

Carrier LI c= I by A49, RLVECT_2:def 6;

then Carrier LI c= I19 by A3;

then A51: LI = L1 by A48, A28, A36, A50, RLVECT_5:1;

Carrier LI c= I by A49, RLVECT_2:def 6;

then ( Carrier (LI + L2) c= (Carrier LI) \/ (Carrier L2) & (Carrier LI) \/ (Carrier L2) c= I2 ) by A46, A32, RLVECT_2:37, XBOOLE_1:13;

then A52: Carrier (LI + L2) c= I2 ;

W2 is Subspace of W1 + W2 by RUSUB_2:7;

then consider K being Linear_Combination of W2 such that

A53: Carrier K = Carrier (LI + L2) and

A54: Sum K = Sum (LI + L2) by A52, RUSUB_3:20, XBOOLE_1:1;

reconsider K = K as Linear_Combination of I2 by A52, A53, RLVECT_2:def 6;

0. W2 = (Sum LI) + (Sum L2) by A28, A35, A36, A50, RUSUB_1:5

.= Sum K by A54, RLVECT_3:1 ;

then {} = Carrier (L1 + L2) by A53, A51, A47, RLVECT_3:def 1;

hence Carrier L = {} by A38; :: thesis: verum

end;then reconsider I19 = I1 as linearly-independent Subset of (W1 + W2) by RUSUB_3:22;

reconsider W29 = W2 as Subspace of W1 + W2 by RUSUB_2:7;

reconsider W19 = W1 as Subspace of W1 + W2 by RUSUB_2:7;

let L be Linear_Combination of A; :: thesis: ( Sum L = 0. (W1 + W2) implies Carrier L = {} )

assume A16: Sum L = 0. (W1 + W2) ; :: thesis: Carrier L = {}

A17: I1 misses (Carrier L) \ I1 by XBOOLE_1:79;

set B = (Carrier L) /\ I1;

consider F being FinSequence of the carrier of (W1 + W2) such that

A18: F is one-to-one and

A19: rng F = Carrier L and

A20: Sum L = Sum (L (#) F) by RLVECT_2:def 8;

reconsider B = (Carrier L) /\ I1 as Subset of (rng F) by A19, XBOOLE_1:17;

reconsider F1 = F - (B `), F2 = F - B as FinSequence of the carrier of (W1 + W2) by FINSEQ_3:86;

consider L1 being Linear_Combination of W1 + W2 such that

A21: Carrier L1 = (rng F1) /\ (Carrier L) and

A22: L1 (#) F1 = L (#) F1 by RLVECT_5:7;

F1 is one-to-one by A18, FINSEQ_3:87;

then A23: Sum (L (#) F1) = Sum L1 by A21, A22, RLVECT_5:6, XBOOLE_1:17;

rng F c= rng F ;

then reconsider X = rng F as Subset of (rng F) ;

consider L2 being Linear_Combination of W1 + W2 such that

A24: Carrier L2 = (rng F2) /\ (Carrier L) and

A25: L2 (#) F2 = L (#) F2 by RLVECT_5:7;

F2 is one-to-one by A18, FINSEQ_3:87;

then A26: Sum (L (#) F2) = Sum L2 by A24, A25, RLVECT_5:6, XBOOLE_1:17;

X \ (B `) = X /\ ((B `) `) by SUBSET_1:13

.= B by XBOOLE_1:28 ;

then rng F1 = B by FINSEQ_3:65;

then A27: Carrier L1 = I1 /\ ((Carrier L) /\ (Carrier L)) by A21, XBOOLE_1:16

.= (Carrier L) /\ I1 ;

then consider K1 being Linear_Combination of W19 such that

Carrier K1 = Carrier L1 and

A28: Sum K1 = Sum L1 by RUSUB_3:20;

rng F2 = (Carrier L) \ ((Carrier L) /\ I1) by A19, FINSEQ_3:65

.= (Carrier L) \ I1 by XBOOLE_1:47 ;

then A29: Carrier L2 = (Carrier L) \ I1 by A24, XBOOLE_1:28, XBOOLE_1:36;

then (Carrier L1) /\ (Carrier L2) = (Carrier L) /\ (I1 /\ ((Carrier L) \ I1)) by A27, XBOOLE_1:16

.= (Carrier L) /\ {} by A17

.= {} ;

then A30: Carrier L1 misses Carrier L2 ;

A31: Carrier L c= I1 \/ I2 by RLVECT_2:def 6;

then A32: Carrier L2 c= I2 by A29, XBOOLE_1:43;

Carrier L2 c= I2 by A31, A29, XBOOLE_1:43;

then consider K2 being Linear_Combination of W29 such that

Carrier K2 = Carrier L2 and

A33: Sum K2 = Sum L2 by RUSUB_3:20, XBOOLE_1:1;

A34: Sum K1 in W1 ;

ex P being Permutation of (dom F) st (F - (B `)) ^ (F - B) = F * P by FINSEQ_3:115;

then A35: 0. (W1 + W2) = Sum (L (#) (F1 ^ F2)) by A16, A20, RLVECT_5:4

.= Sum ((L (#) F1) ^ (L (#) F2)) by RLVECT_3:34

.= (Sum L1) + (Sum L2) by A23, A26, RLVECT_1:41 ;

then Sum L1 = - (Sum L2) by RLVECT_1:def 10

.= - (Sum K2) by A33, RUSUB_1:9 ;

then Sum K1 in W2 by A28;

then Sum K1 in W1 /\ W2 by A34, RUSUB_2:3;

then Sum K1 in Lin I by A1, RUSUB_3:def 2;

then consider KI being Linear_Combination of I such that

A36: Sum K1 = Sum KI by RUSUB_3:1;

A37: Carrier L = (Carrier L1) \/ (Carrier L2) by A27, A29, XBOOLE_1:51;

A38: now :: thesis: Carrier L c= Carrier (L1 + L2)

A46:
I \/ I2 = I2
by A2, XBOOLE_1:12;assume
not Carrier L c= Carrier (L1 + L2)
; :: thesis: contradiction

then consider x being object such that

A39: x in Carrier L and

A40: not x in Carrier (L1 + L2) ;

reconsider x = x as VECTOR of (W1 + W2) by A39;

A41: 0 = (L1 + L2) . x by A40, RLVECT_2:19

.= (L1 . x) + (L2 . x) by RLVECT_2:def 10 ;

end;then consider x being object such that

A39: x in Carrier L and

A40: not x in Carrier (L1 + L2) ;

reconsider x = x as VECTOR of (W1 + W2) by A39;

A41: 0 = (L1 + L2) . x by A40, RLVECT_2:19

.= (L1 . x) + (L2 . x) by RLVECT_2:def 10 ;

per cases
( x in Carrier L1 or x in Carrier L2 )
by A37, A39, XBOOLE_0:def 3;

end;

suppose A42:
x in Carrier L1
; :: thesis: contradiction

then
not x in Carrier L2
by A30, XBOOLE_0:3;

then A43: L2 . x = 0 by RLVECT_2:19;

ex v being VECTOR of (W1 + W2) st

( x = v & L1 . v <> 0 ) by A42, RLVECT_5:3;

hence contradiction by A41, A43; :: thesis: verum

end;then A43: L2 . x = 0 by RLVECT_2:19;

ex v being VECTOR of (W1 + W2) st

( x = v & L1 . v <> 0 ) by A42, RLVECT_5:3;

hence contradiction by A41, A43; :: thesis: verum

suppose A44:
x in Carrier L2
; :: thesis: contradiction

then
not x in Carrier L1
by A30, XBOOLE_0:3;

then A45: L1 . x = 0 by RLVECT_2:19;

ex v being VECTOR of (W1 + W2) st

( x = v & L2 . v <> 0 ) by A44, RLVECT_5:3;

hence contradiction by A41, A45; :: thesis: verum

end;then A45: L1 . x = 0 by RLVECT_2:19;

ex v being VECTOR of (W1 + W2) st

( x = v & L2 . v <> 0 ) by A44, RLVECT_5:3;

hence contradiction by A41, A45; :: thesis: verum

A47: I2 is linearly-independent by RUSUB_3:def 2;

A48: Carrier L1 c= I1 by A27, XBOOLE_1:17;

W1 /\ W2 is Subspace of W1 + W2 by RUSUB_2:22;

then consider LI being Linear_Combination of W1 + W2 such that

A49: Carrier LI = Carrier KI and

A50: Sum LI = Sum KI by RUSUB_3:19;

Carrier LI c= I by A49, RLVECT_2:def 6;

then Carrier LI c= I19 by A3;

then A51: LI = L1 by A48, A28, A36, A50, RLVECT_5:1;

Carrier LI c= I by A49, RLVECT_2:def 6;

then ( Carrier (LI + L2) c= (Carrier LI) \/ (Carrier L2) & (Carrier LI) \/ (Carrier L2) c= I2 ) by A46, A32, RLVECT_2:37, XBOOLE_1:13;

then A52: Carrier (LI + L2) c= I2 ;

W2 is Subspace of W1 + W2 by RUSUB_2:7;

then consider K being Linear_Combination of W2 such that

A53: Carrier K = Carrier (LI + L2) and

A54: Sum K = Sum (LI + L2) by A52, RUSUB_3:20, XBOOLE_1:1;

reconsider K = K as Linear_Combination of I2 by A52, A53, RLVECT_2:def 6;

0. W2 = (Sum LI) + (Sum L2) by A28, A35, A36, A50, RUSUB_1:5

.= Sum K by A54, RLVECT_3:1 ;

then {} = Carrier (L1 + L2) by A53, A51, A47, RLVECT_3:def 1;

hence Carrier L = {} by A38; :: thesis: verum

the carrier of (W1 + W2) c= the carrier of V by RUSUB_1:def 1;

then reconsider A9 = A as Subset of V by XBOOLE_1:1;

A56: card I2 = dim W2 by Def2;

now :: thesis: for x being object st x in the carrier of (W1 + W2) holds

x in the carrier of (Lin A9)

then
the carrier of (W1 + W2) c= the carrier of (Lin A9)
;x in the carrier of (Lin A9)

let x be object ; :: thesis: ( x in the carrier of (W1 + W2) implies x in the carrier of (Lin A9) )

assume x in the carrier of (W1 + W2) ; :: thesis: x in the carrier of (Lin A9)

then x in W1 + W2 ;

then consider w1, w2 being VECTOR of V such that

A57: w1 in W1 and

A58: w2 in W2 and

A59: x = w1 + w2 by RUSUB_2:1;

reconsider w1 = w1 as VECTOR of W1 by A57;

w1 in Lin I1 by RUSUB_3:21;

then consider K1 being Linear_Combination of I1 such that

A60: w1 = Sum K1 by RUSUB_3:1;

reconsider w2 = w2 as VECTOR of W2 by A58;

w2 in Lin I2 by RUSUB_3:21;

then consider K2 being Linear_Combination of I2 such that

A61: w2 = Sum K2 by RUSUB_3:1;

consider L2 being Linear_Combination of V such that

A62: Carrier L2 = Carrier K2 and

A63: Sum L2 = Sum K2 by RUSUB_3:19;

A64: Carrier L2 c= I2 by A62, RLVECT_2:def 6;

consider L1 being Linear_Combination of V such that

A65: Carrier L1 = Carrier K1 and

A66: Sum L1 = Sum K1 by RUSUB_3:19;

set L = L1 + L2;

Carrier L1 c= I1 by A65, RLVECT_2:def 6;

then ( Carrier (L1 + L2) c= (Carrier L1) \/ (Carrier L2) & (Carrier L1) \/ (Carrier L2) c= I1 \/ I2 ) by A64, RLVECT_2:37, XBOOLE_1:13;

then Carrier (L1 + L2) c= I1 \/ I2 ;

then reconsider L = L1 + L2 as Linear_Combination of A9 by RLVECT_2:def 6;

x = Sum L by A59, A60, A66, A61, A63, RLVECT_3:1;

then x in Lin A9 by RUSUB_3:1;

hence x in the carrier of (Lin A9) ; :: thesis: verum

end;assume x in the carrier of (W1 + W2) ; :: thesis: x in the carrier of (Lin A9)

then x in W1 + W2 ;

then consider w1, w2 being VECTOR of V such that

A57: w1 in W1 and

A58: w2 in W2 and

A59: x = w1 + w2 by RUSUB_2:1;

reconsider w1 = w1 as VECTOR of W1 by A57;

w1 in Lin I1 by RUSUB_3:21;

then consider K1 being Linear_Combination of I1 such that

A60: w1 = Sum K1 by RUSUB_3:1;

reconsider w2 = w2 as VECTOR of W2 by A58;

w2 in Lin I2 by RUSUB_3:21;

then consider K2 being Linear_Combination of I2 such that

A61: w2 = Sum K2 by RUSUB_3:1;

consider L2 being Linear_Combination of V such that

A62: Carrier L2 = Carrier K2 and

A63: Sum L2 = Sum K2 by RUSUB_3:19;

A64: Carrier L2 c= I2 by A62, RLVECT_2:def 6;

consider L1 being Linear_Combination of V such that

A65: Carrier L1 = Carrier K1 and

A66: Sum L1 = Sum K1 by RUSUB_3:19;

set L = L1 + L2;

Carrier L1 c= I1 by A65, RLVECT_2:def 6;

then ( Carrier (L1 + L2) c= (Carrier L1) \/ (Carrier L2) & (Carrier L1) \/ (Carrier L2) c= I1 \/ I2 ) by A64, RLVECT_2:37, XBOOLE_1:13;

then Carrier (L1 + L2) c= I1 \/ I2 ;

then reconsider L = L1 + L2 as Linear_Combination of A9 by RLVECT_2:def 6;

x = Sum L by A59, A60, A66, A61, A63, RLVECT_3:1;

then x in Lin A9 by RUSUB_3:1;

hence x in the carrier of (Lin A9) ; :: thesis: verum

then ( Lin A9 = Lin A & W1 + W2 is Subspace of Lin A9 ) by RUSUB_1:22, RUSUB_3:28;

then Lin A = W1 + W2 by RUSUB_1:20;

then A67: A is Basis of W1 + W2 by A55, RUSUB_3:def 2;

card I = dim (W1 /\ W2) by A1, Def2;

then (dim (W1 + W2)) + (dim (W1 /\ W2)) = (((card I1) + (card I2)) + (- (card I))) + (card I) by A15, A67, Def2

.= (dim W1) + (dim W2) by A56, Def2 ;

hence (dim (W1 + W2)) + (dim (W1 /\ W2)) = (dim W1) + (dim W2) ; :: thesis: verum