:: RUSUB_4 semantic presentation begin theorem Th1: :: RUSUB_4:1 for V being RealUnitarySpace for A, B being finite Subset of V for v being VECTOR of V st v in Lin (A \/ B) & not v in Lin B holds ex w being VECTOR of V st ( w in A & w in Lin (((A \/ B) \ {w}) \/ {v}) ) proof let V be RealUnitarySpace; ::_thesis: for A, B being finite Subset of V for v being VECTOR of V st v in Lin (A \/ B) & not v in Lin B holds ex w being VECTOR of V st ( w in A & w in Lin (((A \/ B) \ {w}) \/ {v}) ) let A, B be finite Subset of V; ::_thesis: for v being VECTOR of V st v in Lin (A \/ B) & not v in Lin B holds ex w being VECTOR of V st ( w in A & w in Lin (((A \/ B) \ {w}) \/ {v}) ) let v be VECTOR of V; ::_thesis: ( v in Lin (A \/ B) & not v in Lin B implies ex w being VECTOR of V st ( w in A & w in Lin (((A \/ B) \ {w}) \/ {v}) ) ) assume that A1: v in Lin (A \/ B) and A2: not v in Lin B ; ::_thesis: ex w being VECTOR of V st ( w in A & w in Lin (((A \/ B) \ {w}) \/ {v}) ) consider L being Linear_Combination of A \/ B such that A3: v = Sum L by A1, RUSUB_3:1; v in {v} by TARSKI:def_1; then v in Lin {v} by RUSUB_3:2; then consider Lv being Linear_Combination of {v} such that A4: v = Sum Lv by RUSUB_3:1; A5: Carrier L c= A \/ B by RLVECT_2:def_6; now__::_thesis:_ex_w_being_VECTOR_of_V_st_ (_w_in_A_&_not_L_._w_=_0_) assume A6: for w being VECTOR of V st w in A holds L . w = 0 ; ::_thesis: contradiction now__::_thesis:_for_x_being_set_st_x_in_Carrier_L_holds_ not_x_in_A let x be set ; ::_thesis: ( x in Carrier L implies not x in A ) assume that A7: x in Carrier L and A8: x in A ; ::_thesis: contradiction ex u being VECTOR of V st ( x = u & L . u <> 0 ) by A7, RLVECT_5:3; hence contradiction by A6, A8; ::_thesis: verum end; then Carrier L misses A by XBOOLE_0:3; then Carrier L c= B by A5, XBOOLE_1:73; then L is Linear_Combination of B by RLVECT_2:def_6; hence contradiction by A2, A3, RUSUB_3:1; ::_thesis: verum end; then consider w being VECTOR of V such that A9: w in A and A10: L . w <> 0 ; consider F being FinSequence of the carrier of V such that A11: F is one-to-one and A12: rng F = Carrier L and A13: Sum L = Sum (L (#) F) by RLVECT_2:def_8; A14: w in rng F by A10, A12, RLVECT_5:3; then reconsider Fw1 = F -| w as FinSequence of the carrier of V by FINSEQ_4:41; reconsider Fw2 = F |-- w as FinSequence of the carrier of V by A14, FINSEQ_4:50; A15: rng Fw1 misses rng Fw2 by A11, A14, FINSEQ_4:57; set Fw = Fw1 ^ Fw2; consider K being Linear_Combination of V such that A16: Carrier K = (rng (Fw1 ^ Fw2)) /\ (Carrier L) and A17: L (#) (Fw1 ^ Fw2) = K (#) (Fw1 ^ Fw2) by RLVECT_5:7; F just_once_values w by A11, A14, FINSEQ_4:8; then Fw1 ^ Fw2 = F - {w} by FINSEQ_4:54; then A18: rng (Fw1 ^ Fw2) = (Carrier L) \ {w} by A12, FINSEQ_3:65; then A19: Carrier K = rng (Fw1 ^ Fw2) by A16, XBOOLE_1:28, XBOOLE_1:36; then A20: Carrier K c= (A \/ B) \ {w} by A5, A18, XBOOLE_1:33; take w ; ::_thesis: ( w in A & w in Lin (((A \/ B) \ {w}) \/ {v}) ) set a = L . w; A21: (L . w) " <> 0 by A10, XCMPLX_1:202; F = ((F -| w) ^ <*w*>) ^ (F |-- w) by A14, FINSEQ_4:51; then F = Fw1 ^ (<*w*> ^ Fw2) by FINSEQ_1:32; then L (#) F = (L (#) Fw1) ^ (L (#) (<*w*> ^ Fw2)) by RLVECT_3:34 .= (L (#) Fw1) ^ ((L (#) <*w*>) ^ (L (#) Fw2)) by RLVECT_3:34 .= ((L (#) Fw1) ^ (L (#) <*w*>)) ^ (L (#) Fw2) by FINSEQ_1:32 .= ((L (#) Fw1) ^ <*((L . w) * w)*>) ^ (L (#) Fw2) by RLVECT_2:26 ; then A22: Sum (L (#) F) = Sum ((L (#) Fw1) ^ (<*((L . w) * w)*> ^ (L (#) Fw2))) by FINSEQ_1:32 .= (Sum (L (#) Fw1)) + (Sum (<*((L . w) * w)*> ^ (L (#) Fw2))) by RLVECT_1:41 .= (Sum (L (#) Fw1)) + ((Sum <*((L . w) * w)*>) + (Sum (L (#) Fw2))) by RLVECT_1:41 .= (Sum (L (#) Fw1)) + ((Sum (L (#) Fw2)) + ((L . w) * w)) by RLVECT_1:44 .= ((Sum (L (#) Fw1)) + (Sum (L (#) Fw2))) + ((L . w) * w) by RLVECT_1:def_3 .= (Sum ((L (#) Fw1) ^ (L (#) Fw2))) + ((L . w) * w) by RLVECT_1:41 .= (Sum (L (#) (Fw1 ^ Fw2))) + ((L . w) * w) by RLVECT_3:34 ; reconsider K = K as Linear_Combination of (A \/ B) \ {w} by A20, RLVECT_2:def_6; Carrier ((- K) + Lv) c= (Carrier (- K)) \/ (Carrier Lv) by RLVECT_2:37; then A23: Carrier ((- K) + Lv) c= (Carrier K) \/ (Carrier Lv) by RLVECT_2:51; set LC = ((L . w) ") * ((- K) + Lv); Carrier Lv c= {v} by RLVECT_2:def_6; then (Carrier K) \/ (Carrier Lv) c= ((A \/ B) \ {w}) \/ {v} by A20, XBOOLE_1:13; then Carrier ((- K) + Lv) c= ((A \/ B) \ {w}) \/ {v} by A23, XBOOLE_1:1; then Carrier (((L . w) ") * ((- K) + Lv)) c= ((A \/ B) \ {w}) \/ {v} by A21, RLVECT_2:42; then A24: ((L . w) ") * ((- K) + Lv) is Linear_Combination of ((A \/ B) \ {w}) \/ {v} by RLVECT_2:def_6; ( Fw1 is one-to-one & Fw2 is one-to-one ) by A11, A14, FINSEQ_4:52, FINSEQ_4:53; then Fw1 ^ Fw2 is one-to-one by A15, FINSEQ_3:91; then Sum (K (#) (Fw1 ^ Fw2)) = Sum K by A19, RLVECT_2:def_8; then ((L . w) ") * v = (((L . w) ") * (Sum K)) + (((L . w) ") * ((L . w) * w)) by A3, A13, A22, A17, RLVECT_1:def_5 .= (((L . w) ") * (Sum K)) + ((((L . w) ") * (L . w)) * w) by RLVECT_1:def_7 .= (((L . w) ") * (Sum K)) + (1 * w) by A10, XCMPLX_0:def_7 .= (((L . w) ") * (Sum K)) + w by RLVECT_1:def_8 ; then w = (((L . w) ") * v) - (((L . w) ") * (Sum K)) by RLSUB_2:61 .= ((L . w) ") * (v - (Sum K)) by RLVECT_1:34 .= ((L . w) ") * ((- (Sum K)) + v) by RLVECT_1:def_11 ; then w = ((L . w) ") * ((Sum (- K)) + (Sum Lv)) by A4, RLVECT_3:3 .= ((L . w) ") * (Sum ((- K) + Lv)) by RLVECT_3:1 .= Sum (((L . w) ") * ((- K) + Lv)) by RLVECT_3:2 ; hence ( w in A & w in Lin (((A \/ B) \ {w}) \/ {v}) ) by A9, A24, RUSUB_3:1; ::_thesis: verum end; Lm1: for X, x being set st x in X holds (X \ {x}) \/ {x} = X proof let X, x be set ; ::_thesis: ( x in X implies (X \ {x}) \/ {x} = X ) assume x in X ; ::_thesis: (X \ {x}) \/ {x} = X then A1: {x} is Subset of X by SUBSET_1:41; {x} \/ (X \ {x}) = {x} \/ X by XBOOLE_1:39 .= X by A1, XBOOLE_1:12 ; hence (X \ {x}) \/ {x} = X ; ::_thesis: verum end; Lm2: for X, x being set st not x in X holds X \ {x} = X proof let X, x be set ; ::_thesis: ( not x in X implies X \ {x} = X ) assume A1: not x in X ; ::_thesis: X \ {x} = X now__::_thesis:_not_X_meets_{x} assume X meets {x} ; ::_thesis: contradiction then consider y being set such that A2: y in X /\ {x} by XBOOLE_0:4; ( y in X & y in {x} ) by A2, XBOOLE_0:def_4; hence contradiction by A1, TARSKI:def_1; ::_thesis: verum end; hence X \ {x} = X by XBOOLE_1:83; ::_thesis: verum end; theorem Th2: :: RUSUB_4:2 for V being RealUnitarySpace for A, B being finite Subset of V st UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin A & B is linearly-independent holds ( card B <= card A & ex C being finite Subset of V st ( C c= A & card C = (card A) - (card B) & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin (B \/ C) ) ) proof let V be RealUnitarySpace; ::_thesis: for A, B being finite Subset of V st UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin A & B is linearly-independent holds ( card B <= card A & ex C being finite Subset of V st ( C c= A & card C = (card A) - (card B) & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin (B \/ C) ) ) defpred S1[ Element of NAT ] means for n being Element of NAT for A, B being finite Subset of V st card A = n & card B = $1 & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin A & B is linearly-independent holds ( $1 <= n & ex C being finite Subset of V st ( C c= A & card C = n - $1 & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin (B \/ C) ) ); A1: for m being Element of NAT st S1[m] holds S1[m + 1] proof let m be Element of NAT ; ::_thesis: ( S1[m] implies S1[m + 1] ) assume A2: S1[m] ; ::_thesis: S1[m + 1] let n be Element of NAT ; ::_thesis: for A, B being finite Subset of V st card A = n & card B = m + 1 & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin A & B is linearly-independent holds ( m + 1 <= n & ex C being finite Subset of V st ( C c= A & card C = n - (m + 1) & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin (B \/ C) ) ) let A, B be finite Subset of V; ::_thesis: ( card A = n & card B = m + 1 & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin A & B is linearly-independent implies ( m + 1 <= n & ex C being finite Subset of V st ( C c= A & card C = n - (m + 1) & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin (B \/ C) ) ) ) assume that A3: card A = n and A4: card B = m + 1 and A5: UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin A and A6: B is linearly-independent ; ::_thesis: ( m + 1 <= n & ex C being finite Subset of V st ( C c= A & card C = n - (m + 1) & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin (B \/ C) ) ) consider v being set such that A7: v in B by A4, CARD_1:27, XBOOLE_0:def_1; reconsider v = v as VECTOR of V by A7; set Bv = B \ {v}; A8: B \ {v} is linearly-independent by A6, RLVECT_3:5, XBOOLE_1:36; {v} is Subset of B by A7, SUBSET_1:41; then A9: card (B \ {v}) = (card B) - (card {v}) by CARD_2:44 .= (m + 1) - 1 by A4, CARD_1:30 .= m ; then consider C being finite Subset of V such that A10: C c= A and A11: card C = n - m and A12: UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin ((B \ {v}) \/ C) by A2, A3, A5, A8; A13: not v in Lin (B \ {v}) by A6, A7, RUSUB_3:25; A14: now__::_thesis:_not_m_=_n assume m = n ; ::_thesis: contradiction then consider C being finite Subset of V such that C c= A and A15: card C = m - m and A16: UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin ((B \ {v}) \/ C) by A2, A3, A5, A9, A8; C = {} by A15; then B \ {v} is Basis of V by A8, A16, RUSUB_3:def_2; hence contradiction by A13, RUSUB_3:21; ::_thesis: verum end; v in Lin ((B \ {v}) \/ C) by A12, STRUCT_0:def_5; then consider w being VECTOR of V such that A17: w in C and A18: w in Lin (((C \/ (B \ {v})) \ {w}) \/ {v}) by A13, Th1; set Cw = C \ {w}; ((B \ {v}) \ {w}) \/ {v} c= (B \ {v}) \/ {v} by XBOOLE_1:9, XBOOLE_1:36; then (C \ {w}) \/ (((B \ {v}) \ {w}) \/ {v}) c= (C \ {w}) \/ ((B \ {v}) \/ {v}) by XBOOLE_1:9; then A19: (C \ {w}) \/ (((B \ {v}) \ {w}) \/ {v}) c= B \/ (C \ {w}) by A7, Lm1; {w} is Subset of C by A17, SUBSET_1:41; then A20: card (C \ {w}) = (card C) - (card {w}) by CARD_2:44 .= (n - m) - 1 by A11, CARD_1:30 .= n - (m + 1) ; C \ {w} c= C by XBOOLE_1:36; then A21: C \ {w} c= A by A10, XBOOLE_1:1; ((C \/ (B \ {v})) \ {w}) \/ {v} = ((C \ {w}) \/ ((B \ {v}) \ {w})) \/ {v} by XBOOLE_1:42 .= (C \ {w}) \/ (((B \ {v}) \ {w}) \/ {v}) by XBOOLE_1:4 ; then Lin (((C \/ (B \ {v})) \ {w}) \/ {v}) is Subspace of Lin (B \/ (C \ {w})) by A19, RUSUB_3:7; then A22: w in Lin (B \/ (C \ {w})) by A18, RUSUB_1:1; A23: ( B \ {v} c= B & C = (C \ {w}) \/ {w} ) by A17, Lm1, XBOOLE_1:36; now__::_thesis:_for_x_being_set_st_x_in_(B_\_{v})_\/_C_holds_ x_in_the_carrier_of_(Lin_(B_\/_(C_\_{w}))) let x be set ; ::_thesis: ( x in (B \ {v}) \/ C implies x in the carrier of (Lin (B \/ (C \ {w}))) ) assume x in (B \ {v}) \/ C ; ::_thesis: x in the carrier of (Lin (B \/ (C \ {w}))) then ( x in B \ {v} or x in C ) by XBOOLE_0:def_3; then ( x in B or x in C \ {w} or x in {w} ) by A23, XBOOLE_0:def_3; then ( x in B \/ (C \ {w}) or x in {w} ) by XBOOLE_0:def_3; then ( x in Lin (B \/ (C \ {w})) or x = w ) by RUSUB_3:2, TARSKI:def_1; hence x in the carrier of (Lin (B \/ (C \ {w}))) by A22, STRUCT_0:def_5; ::_thesis: verum end; then (B \ {v}) \/ C c= the carrier of (Lin (B \/ (C \ {w}))) by TARSKI:def_3; then Lin ((B \ {v}) \/ C) is Subspace of Lin (B \/ (C \ {w})) by RUSUB_3:27; then A24: the carrier of (Lin ((B \ {v}) \/ C)) c= the carrier of (Lin (B \/ (C \ {w}))) by RUSUB_1:def_1; the carrier of (Lin (B \/ (C \ {w}))) c= the carrier of V by RUSUB_1:def_1; then the carrier of (Lin (B \/ (C \ {w}))) = the carrier of V by A12, A24, XBOOLE_0:def_10; then A25: UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin (B \/ (C \ {w})) by A12, RUSUB_1:24; m <= n by A2, A3, A5, A9, A8; then m < n by A14, XXREAL_0:1; hence ( m + 1 <= n & ex C being finite Subset of V st ( C c= A & card C = n - (m + 1) & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin (B \/ C) ) ) by A21, A20, A25, NAT_1:13; ::_thesis: verum end; A26: S1[ 0 ] proof let n be Element of NAT ; ::_thesis: for A, B being finite Subset of V st card A = n & card B = 0 & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin A & B is linearly-independent holds ( 0 <= n & ex C being finite Subset of V st ( C c= A & card C = n - 0 & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin (B \/ C) ) ) let A, B be finite Subset of V; ::_thesis: ( card A = n & card B = 0 & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin A & B is linearly-independent implies ( 0 <= n & ex C being finite Subset of V st ( C c= A & card C = n - 0 & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin (B \/ C) ) ) ) assume that A27: card A = n and A28: card B = 0 and A29: UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin A and B is linearly-independent ; ::_thesis: ( 0 <= n & ex C being finite Subset of V st ( C c= A & card C = n - 0 & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin (B \/ C) ) ) B = {} by A28; then A = B \/ A ; hence ( 0 <= n & ex C being finite Subset of V st ( C c= A & card C = n - 0 & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin (B \/ C) ) ) by A27, A29, NAT_1:2; ::_thesis: verum end; A30: for m being Element of NAT holds S1[m] from NAT_1:sch_1(A26, A1); let A, B be finite Subset of V; ::_thesis: ( UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin A & B is linearly-independent implies ( card B <= card A & ex C being finite Subset of V st ( C c= A & card C = (card A) - (card B) & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin (B \/ C) ) ) ) assume ( UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin A & B is linearly-independent ) ; ::_thesis: ( card B <= card A & ex C being finite Subset of V st ( C c= A & card C = (card A) - (card B) & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin (B \/ C) ) ) hence ( card B <= card A & ex C being finite Subset of V st ( C c= A & card C = (card A) - (card B) & UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin (B \/ C) ) ) by A30; ::_thesis: verum end; definition let V be RealUnitarySpace; attrV is finite-dimensional means :Def1: :: RUSUB_4:def 1 ex A being finite Subset of V st A is Basis of V; end; :: deftheorem Def1 defines finite-dimensional RUSUB_4:def_1_:_ for V being RealUnitarySpace holds ( V is finite-dimensional iff ex A being finite Subset of V st A is Basis of V ); registration cluster non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V115() strict RealUnitarySpace-like finite-dimensional for UNITSTR ; existence ex b1 being RealUnitarySpace st ( b1 is strict & b1 is finite-dimensional ) proof set V = the RealUnitarySpace; take (0). the RealUnitarySpace ; ::_thesis: ( (0). the RealUnitarySpace is strict & (0). the RealUnitarySpace is finite-dimensional ) thus (0). the RealUnitarySpace is strict ; ::_thesis: (0). the RealUnitarySpace is finite-dimensional take A = {} the carrier of ((0). the RealUnitarySpace); :: according to RUSUB_4:def_1 ::_thesis: A is Basis of (0). the RealUnitarySpace Lin A = (0). ((0). the RealUnitarySpace) by RUSUB_3:3; then ( A is linearly-independent & Lin A = UNITSTR(# the carrier of ((0). the RealUnitarySpace), the ZeroF of ((0). the RealUnitarySpace), the addF of ((0). the RealUnitarySpace), the Mult of ((0). the RealUnitarySpace), the scalar of ((0). the RealUnitarySpace) #) ) by RLVECT_3:7, RUSUB_1:30; hence A is Basis of (0). the RealUnitarySpace by RUSUB_3:def_2; ::_thesis: verum end; end; theorem Th3: :: RUSUB_4:3 for V being RealUnitarySpace st V is finite-dimensional holds for I being Basis of V holds I is finite proof let V be RealUnitarySpace; ::_thesis: ( V is finite-dimensional implies for I being Basis of V holds I is finite ) assume V is finite-dimensional ; ::_thesis: for I being Basis of V holds I is finite then consider A being finite Subset of V such that A1: A is Basis of V by Def1; let B be Basis of V; ::_thesis: B is finite consider p being FinSequence such that A2: rng p = A by FINSEQ_1:52; reconsider p = p as FinSequence of the carrier of V by A2, FINSEQ_1:def_4; set Car = { (Carrier L) where L is Linear_Combination of B : ex i being Element of NAT st ( i in dom p & Sum L = p . i ) } ; set C = union { (Carrier L) where L is Linear_Combination of B : ex i being Element of NAT st ( i in dom p & Sum L = p . i ) } ; A3: union { (Carrier L) where L is Linear_Combination of B : ex i being Element of NAT st ( i in dom p & Sum L = p . i ) } c= B proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union { (Carrier L) where L is Linear_Combination of B : ex i being Element of NAT st ( i in dom p & Sum L = p . i ) } or x in B ) assume x in union { (Carrier L) where L is Linear_Combination of B : ex i being Element of NAT st ( i in dom p & Sum L = p . i ) } ; ::_thesis: x in B then consider R being set such that A4: x in R and A5: R in { (Carrier L) where L is Linear_Combination of B : ex i being Element of NAT st ( i in dom p & Sum L = p . i ) } by TARSKI:def_4; ex L being Linear_Combination of B st ( R = Carrier L & ex i being Element of NAT st ( i in dom p & Sum L = p . i ) ) by A5; then R c= B by RLVECT_2:def_6; hence x in B by A4; ::_thesis: verum end; then reconsider C = union { (Carrier L) where L is Linear_Combination of B : ex i being Element of NAT st ( i in dom p & Sum L = p . i ) } as Subset of V by XBOOLE_1:1; for v being VECTOR of V holds ( v in (Omega). V iff v in Lin C ) proof let v be VECTOR of V; ::_thesis: ( v in (Omega). V iff v in Lin C ) hereby ::_thesis: ( v in Lin C implies v in (Omega). V ) assume v in (Omega). V ; ::_thesis: v in Lin C then v in UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) by RUSUB_1:def_3; then v in Lin A by A1, RUSUB_3:def_2; then consider LA being Linear_Combination of A such that A6: v = Sum LA by RUSUB_3:1; Carrier LA c= the carrier of (Lin C) proof let w be set ; :: according to TARSKI:def_3 ::_thesis: ( not w in Carrier LA or w in the carrier of (Lin C) ) assume A7: w in Carrier LA ; ::_thesis: w in the carrier of (Lin C) then reconsider w9 = w as VECTOR of V ; w9 in Lin B by RUSUB_3:21; then consider LB being Linear_Combination of B such that A8: w = Sum LB by RUSUB_3:1; Carrier LA c= A by RLVECT_2:def_6; then ex i being set st ( i in dom p & w = p . i ) by A2, A7, FUNCT_1:def_3; then A9: Carrier LB in { (Carrier L) where L is Linear_Combination of B : ex i being Element of NAT st ( i in dom p & Sum L = p . i ) } by A8; Carrier LB c= C proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Carrier LB or x in C ) assume x in Carrier LB ; ::_thesis: x in C hence x in C by A9, TARSKI:def_4; ::_thesis: verum end; then LB is Linear_Combination of C by RLVECT_2:def_6; then w in Lin C by A8, RUSUB_3:1; hence w in the carrier of (Lin C) by STRUCT_0:def_5; ::_thesis: verum end; then ex LC being Linear_Combination of C st Sum LA = Sum LC by RUSUB_3:17; hence v in Lin C by A6, RUSUB_3:1; ::_thesis: verum end; assume v in Lin C ; ::_thesis: v in (Omega). V v in the carrier of UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) ; then v in the carrier of ((Omega). V) by RUSUB_1:def_3; hence v in (Omega). V by STRUCT_0:def_5; ::_thesis: verum end; then (Omega). V = Lin C by RUSUB_1:25; then A10: UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin C by RUSUB_1:def_3; A11: B is linearly-independent by RUSUB_3:def_2; then C is linearly-independent by A3, RLVECT_3:5; then A12: C is Basis of V by A10, RUSUB_3:def_2; B c= C proof set D = B \ C; assume not B c= C ; ::_thesis: contradiction then ex v being set st ( v in B & not v in C ) by TARSKI:def_3; then reconsider D = B \ C as non empty Subset of V by XBOOLE_0:def_5; reconsider B = B as Subset of V ; C \/ (B \ C) = C \/ B by XBOOLE_1:39 .= B by A3, XBOOLE_1:12 ; then B = C \/ D ; hence contradiction by A11, A12, RUSUB_3:26, XBOOLE_1:79; ::_thesis: verum end; then A13: B = C by A3, XBOOLE_0:def_10; defpred S1[ set , set ] means ex L being Linear_Combination of B st ( $2 = Carrier L & Sum L = p . $1 ); A14: for i being Nat st i in Seg (len p) holds ex x being set st S1[i,x] proof let i be Nat; ::_thesis: ( i in Seg (len p) implies ex x being set st S1[i,x] ) assume i in Seg (len p) ; ::_thesis: ex x being set st S1[i,x] then i in dom p by FINSEQ_1:def_3; then p . i in the carrier of V by FINSEQ_2:11; then p . i in Lin B by RUSUB_3:21; then consider L being Linear_Combination of B such that A15: p . i = Sum L by RUSUB_3:1; S1[i, Carrier L] by A15; hence ex x being set st S1[i,x] ; ::_thesis: verum end; ex q being FinSequence st ( dom q = Seg (len p) & ( for i being Nat st i in Seg (len p) holds S1[i,q . i] ) ) from FINSEQ_1:sch_1(A14); then consider q being FinSequence such that A16: dom q = Seg (len p) and A17: for i being Nat st i in Seg (len p) holds S1[i,q . i] ; A18: dom p = dom q by A16, FINSEQ_1:def_3; A19: for i being Nat for y1, y2 being set st i in Seg (len p) & S1[i,y1] & S1[i,y2] holds y1 = y2 proof let i be Nat; ::_thesis: for y1, y2 being set st i in Seg (len p) & S1[i,y1] & S1[i,y2] holds y1 = y2 let y1, y2 be set ; ::_thesis: ( i in Seg (len p) & S1[i,y1] & S1[i,y2] implies y1 = y2 ) assume that i in Seg (len p) and A20: S1[i,y1] and A21: S1[i,y2] ; ::_thesis: y1 = y2 consider L1 being Linear_Combination of B such that A22: ( y1 = Carrier L1 & Sum L1 = p . i ) by A20; consider L2 being Linear_Combination of B such that A23: ( y2 = Carrier L2 & Sum L2 = p . i ) by A21; ( Carrier L1 c= B & Carrier L2 c= B ) by RLVECT_2:def_6; hence y1 = y2 by A11, A22, A23, RLVECT_5:1; ::_thesis: verum end; now__::_thesis:_for_x_being_set_st_x_in__{__(Carrier_L)_where_L_is_Linear_Combination_of_B_:_ex_i_being_Element_of_NAT_st_ (_i_in_dom_p_&_Sum_L_=_p_._i_)__}__holds_ x_in_rng_q let x be set ; ::_thesis: ( x in { (Carrier L) where L is Linear_Combination of B : ex i being Element of NAT st ( i in dom p & Sum L = p . i ) } implies x in rng q ) assume x in { (Carrier L) where L is Linear_Combination of B : ex i being Element of NAT st ( i in dom p & Sum L = p . i ) } ; ::_thesis: x in rng q then consider L being Linear_Combination of B such that A24: x = Carrier L and A25: ex i being Element of NAT st ( i in dom p & Sum L = p . i ) ; consider i being Element of NAT such that A26: i in dom p and A27: Sum L = p . i by A25; S1[i,q . i] by A16, A17, A18, A26; then x = q . i by A19, A16, A18, A24, A26, A27; hence x in rng q by A18, A26, FUNCT_1:def_3; ::_thesis: verum end; then A28: { (Carrier L) where L is Linear_Combination of B : ex i being Element of NAT st ( i in dom p & Sum L = p . i ) } c= rng q by TARSKI:def_3; for R being set st R in { (Carrier L) where L is Linear_Combination of B : ex i being Element of NAT st ( i in dom p & Sum L = p . i ) } holds R is finite proof let R be set ; ::_thesis: ( R in { (Carrier L) where L is Linear_Combination of B : ex i being Element of NAT st ( i in dom p & Sum L = p . i ) } implies R is finite ) assume R in { (Carrier L) where L is Linear_Combination of B : ex i being Element of NAT st ( i in dom p & Sum L = p . i ) } ; ::_thesis: R is finite then ex L being Linear_Combination of B st ( R = Carrier L & ex i being Element of NAT st ( i in dom p & Sum L = p . i ) ) ; hence R is finite ; ::_thesis: verum end; hence B is finite by A13, A28, FINSET_1:7; ::_thesis: verum end; theorem :: RUSUB_4:4 for V being RealUnitarySpace for A being Subset of V st V is finite-dimensional & A is linearly-independent holds A is finite proof let V be RealUnitarySpace; ::_thesis: for A being Subset of V st V is finite-dimensional & A is linearly-independent holds A is finite let A be Subset of V; ::_thesis: ( V is finite-dimensional & A is linearly-independent implies A is finite ) assume that A1: V is finite-dimensional and A2: A is linearly-independent ; ::_thesis: A is finite consider B being Basis of V such that A3: A c= B by A2, RUSUB_3:15; B is finite by A1, Th3; hence A is finite by A3; ::_thesis: verum end; theorem Th5: :: RUSUB_4:5 for V being RealUnitarySpace for A, B being Basis of V st V is finite-dimensional holds card A = card B proof let V be RealUnitarySpace; ::_thesis: for A, B being Basis of V st V is finite-dimensional holds card A = card B let A, B be Basis of V; ::_thesis: ( V is finite-dimensional implies card A = card B ) assume V is finite-dimensional ; ::_thesis: card A = card B then reconsider A9 = A, B9 = B as finite Subset of V by Th3; ( UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin B & A9 is linearly-independent ) by RUSUB_3:def_2; then A1: card A9 <= card B9 by Th2; ( UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = Lin A & B9 is linearly-independent ) by RUSUB_3:def_2; then card B9 <= card A9 by Th2; hence card A = card B by A1, XXREAL_0:1; ::_thesis: verum end; theorem Th6: :: RUSUB_4:6 for V being RealUnitarySpace holds (0). V is finite-dimensional proof let V be RealUnitarySpace; ::_thesis: (0). V is finite-dimensional reconsider V9 = (0). V as strict RealUnitarySpace ; reconsider I = {} the carrier of V9 as finite Subset of V9 ; the carrier of V9 = {(0. V)} by RUSUB_1:def_2 .= {(0. V9)} by RUSUB_1:4 .= the carrier of ((0). V9) by RUSUB_1:def_2 ; then A1: V9 = (0). V9 by RUSUB_1:26; ( I is linearly-independent & Lin I = (0). V9 ) by RLVECT_3:7, RUSUB_3:3; then I is Basis of V9 by A1, RUSUB_3:def_2; hence (0). V is finite-dimensional by Def1; ::_thesis: verum end; theorem Th7: :: RUSUB_4:7 for V being RealUnitarySpace for W being Subspace of V st V is finite-dimensional holds W is finite-dimensional proof let V be RealUnitarySpace; ::_thesis: for W being Subspace of V st V is finite-dimensional holds W is finite-dimensional let W be Subspace of V; ::_thesis: ( V is finite-dimensional implies W is finite-dimensional ) set A = the Basis of W; consider I being Basis of V such that A1: the Basis of W c= I by RUSUB_3:24; assume V is finite-dimensional ; ::_thesis: W is finite-dimensional then I is finite by Th3; hence W is finite-dimensional by A1, Def1; ::_thesis: verum end; registration let V be RealUnitarySpace; cluster non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V115() strict RealUnitarySpace-like finite-dimensional for Subspace of V; existence ex b1 being Subspace of V st ( b1 is finite-dimensional & b1 is strict ) proof take (0). V ; ::_thesis: ( (0). V is finite-dimensional & (0). V is strict ) thus ( (0). V is finite-dimensional & (0). V is strict ) by Th6; ::_thesis: verum end; end; registration let V be finite-dimensional RealUnitarySpace; cluster -> finite-dimensional for Subspace of V; correctness coherence for b1 being Subspace of V holds b1 is finite-dimensional ; by Th7; end; registration let V be finite-dimensional RealUnitarySpace; cluster non empty left_complementable right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V115() strict RealUnitarySpace-like finite-dimensional for Subspace of V; existence ex b1 being Subspace of V st b1 is strict proof (0). V is strict finite-dimensional Subspace of V ; hence ex b1 being Subspace of V st b1 is strict ; ::_thesis: verum end; end; begin definition let V be RealUnitarySpace; assume A1: V is finite-dimensional ; func dim V -> Element of NAT means :Def2: :: RUSUB_4:def 2 for I being Basis of V holds it = card I; existence ex b1 being Element of NAT st for I being Basis of V holds b1 = card I proof consider A being finite Subset of V such that A2: A is Basis of V by A1, Def1; consider n being Element of NAT such that A3: n = card A ; for I being Basis of V holds card I = n by A1, A2, A3, Th5; hence ex b1 being Element of NAT st for I being Basis of V holds b1 = card I ; ::_thesis: verum end; uniqueness for b1, b2 being Element of NAT st ( for I being Basis of V holds b1 = card I ) & ( for I being Basis of V holds b2 = card I ) holds b1 = b2 proof let n, m be Element of NAT ; ::_thesis: ( ( for I being Basis of V holds n = card I ) & ( for I being Basis of V holds m = card I ) implies n = m ) assume that A4: for I being Basis of V holds card I = n and A5: for I being Basis of V holds card I = m ; ::_thesis: n = m consider A being finite Subset of V such that A6: A is Basis of V by A1, Def1; card A = n by A4, A6; hence n = m by A5, A6; ::_thesis: verum end; end; :: deftheorem Def2 defines dim RUSUB_4:def_2_:_ for V being RealUnitarySpace st V is finite-dimensional holds for b2 being Element of NAT holds ( b2 = dim V iff for I being Basis of V holds b2 = card I ); theorem Th8: :: RUSUB_4:8 for V being finite-dimensional RealUnitarySpace for W being Subspace of V holds dim W <= dim V proof let V be finite-dimensional RealUnitarySpace; ::_thesis: for W being Subspace of V holds dim W <= dim V let W be Subspace of V; ::_thesis: dim W <= dim V set A = the Basis of W; reconsider A = the Basis of W as Subset of W ; A is linearly-independent by RUSUB_3:def_2; then reconsider B = A as linearly-independent Subset of V by RUSUB_3:22; reconsider A9 = B as finite Subset of V by Th3; reconsider V9 = V as RealUnitarySpace ; set I = the Basis of V9; A1: Lin the Basis of V9 = UNITSTR(# the carrier of V9, the ZeroF of V9, the addF of V9, the Mult of V9, the scalar of V9 #) by RUSUB_3:def_2; reconsider I = the Basis of V9 as finite Subset of V by Th3; A2: dim V = card I by Def2; card A9 <= card I by A1, Th2; hence dim W <= dim V by A2, Def2; ::_thesis: verum end; theorem Th9: :: RUSUB_4:9 for V being finite-dimensional RealUnitarySpace for A being Subset of V st A is linearly-independent holds card A = dim (Lin A) proof let V be finite-dimensional RealUnitarySpace; ::_thesis: for A being Subset of V st A is linearly-independent holds card A = dim (Lin A) let A be Subset of V; ::_thesis: ( A is linearly-independent implies card A = dim (Lin A) ) assume A1: A is linearly-independent ; ::_thesis: card A = dim (Lin A) set W = Lin A; now__::_thesis:_for_x_being_set_st_x_in_A_holds_ x_in_the_carrier_of_(Lin_A) let x be set ; ::_thesis: ( x in A implies x in the carrier of (Lin A) ) assume x in A ; ::_thesis: x in the carrier of (Lin A) then x in Lin A by RUSUB_3:2; hence x in the carrier of (Lin A) by STRUCT_0:def_5; ::_thesis: verum end; then reconsider B = A as linearly-independent Subset of (Lin A) by A1, RUSUB_3:23, TARSKI:def_3; Lin A = Lin B by RUSUB_3:28; then reconsider B = B as Basis of Lin A by RUSUB_3:def_2; card B = dim (Lin A) by Def2; hence card A = dim (Lin A) ; ::_thesis: verum end; theorem Th10: :: RUSUB_4:10 for V being finite-dimensional RealUnitarySpace holds dim V = dim ((Omega). V) proof let V be finite-dimensional RealUnitarySpace; ::_thesis: dim V = dim ((Omega). V) consider I being finite Subset of V such that A1: I is Basis of V by Def1; A2: (Omega). V = UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) by RUSUB_1:def_3 .= Lin I by A1, RUSUB_3:def_2 ; ( card I = dim V & I is linearly-independent ) by A1, Def2, RUSUB_3:def_2; hence dim V = dim ((Omega). V) by A2, Th9; ::_thesis: verum end; theorem :: RUSUB_4:11 for V being finite-dimensional RealUnitarySpace for W being Subspace of V holds ( dim V = dim W iff (Omega). V = (Omega). W ) proof let V be finite-dimensional RealUnitarySpace; ::_thesis: for W being Subspace of V holds ( dim V = dim W iff (Omega). V = (Omega). W ) let W be Subspace of V; ::_thesis: ( dim V = dim W iff (Omega). V = (Omega). W ) consider A being finite Subset of V such that A1: A is Basis of V by Def1; hereby ::_thesis: ( (Omega). V = (Omega). W implies dim V = dim W ) set A = the Basis of W; consider B being Basis of V such that A2: the Basis of W c= B by RUSUB_3:24; the carrier of W c= the carrier of V by RUSUB_1:def_1; then reconsider A9 = the Basis of W as finite Subset of V by Th3, XBOOLE_1:1; reconsider B9 = B as finite Subset of V by Th3; assume dim V = dim W ; ::_thesis: (Omega). V = (Omega). W then A3: card the Basis of W = dim V by Def2 .= card B by Def2 ; A4: now__::_thesis:_not_the_Basis_of_W_<>_B assume the Basis of W <> B ; ::_thesis: contradiction then the Basis of W c< B by A2, XBOOLE_0:def_8; then card A9 < card B9 by CARD_2:48; hence contradiction by A3; ::_thesis: verum end; reconsider B = B as Subset of V ; reconsider A = the Basis of W as Subset of W ; (Omega). V = UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) by RUSUB_1:def_3 .= Lin B by RUSUB_3:def_2 .= Lin A by A4, RUSUB_3:28 .= UNITSTR(# the carrier of W, the ZeroF of W, the addF of W, the Mult of W, the scalar of W #) by RUSUB_3:def_2 .= (Omega). W by RUSUB_1:def_3 ; hence (Omega). V = (Omega). W ; ::_thesis: verum end; consider B being finite Subset of W such that A5: B is Basis of W by Def1; A6: A is linearly-independent by A1, RUSUB_3:def_2; assume (Omega). V = (Omega). W ; ::_thesis: dim V = dim W then UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = (Omega). W by RUSUB_1:def_3 .= UNITSTR(# the carrier of W, the ZeroF of W, the addF of W, the Mult of W, the scalar of W #) by RUSUB_1:def_3 ; then A7: Lin A = UNITSTR(# the carrier of W, the ZeroF of W, the addF of W, the Mult of W, the scalar of W #) by A1, RUSUB_3:def_2 .= Lin B by A5, RUSUB_3:def_2 ; A8: B is linearly-independent by A5, RUSUB_3:def_2; reconsider B = B as Subset of W ; reconsider A = A as Subset of V ; dim V = card A by A1, Def2 .= dim (Lin B) by A6, A7, Th9 .= card B by A8, Th9 .= dim W by A5, Def2 ; hence dim V = dim W ; ::_thesis: verum end; theorem Th12: :: RUSUB_4:12 for V being finite-dimensional RealUnitarySpace holds ( dim V = 0 iff (Omega). V = (0). V ) proof let V be finite-dimensional RealUnitarySpace; ::_thesis: ( dim V = 0 iff (Omega). V = (0). V ) consider I being finite Subset of V such that A1: I is Basis of V by Def1; hereby ::_thesis: ( (Omega). V = (0). V implies dim V = 0 ) consider I being finite Subset of V such that A2: I is Basis of V by Def1; assume dim V = 0 ; ::_thesis: (Omega). V = (0). V then card I = 0 by A2, Def2; then A3: I = {} the carrier of V ; (Omega). V = UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) by RUSUB_1:def_3 .= Lin I by A2, RUSUB_3:def_2 .= (0). V by A3, RUSUB_3:3 ; hence (Omega). V = (0). V ; ::_thesis: verum end; A4: now__::_thesis:_not_I_=_{(0._V)} assume I = {(0. V)} ; ::_thesis: contradiction then I is linearly-dependent by RLVECT_3:8; hence contradiction by A1, RUSUB_3:def_2; ::_thesis: verum end; assume (Omega). V = (0). V ; ::_thesis: dim V = 0 then UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = (0). V by RUSUB_1:def_3; then Lin I = (0). V by A1, RUSUB_3:def_2; then ( I = {} or I = {(0. V)} ) by RUSUB_3:4; hence dim V = 0 by A1, A4, Def2, CARD_1:27; ::_thesis: verum end; theorem :: RUSUB_4:13 for V being finite-dimensional RealUnitarySpace holds ( dim V = 1 iff ex v being VECTOR of V st ( v <> 0. V & (Omega). V = Lin {v} ) ) proof let V be finite-dimensional RealUnitarySpace; ::_thesis: ( dim V = 1 iff ex v being VECTOR of V st ( v <> 0. V & (Omega). V = Lin {v} ) ) hereby ::_thesis: ( ex v being VECTOR of V st ( v <> 0. V & (Omega). V = Lin {v} ) implies dim V = 1 ) consider I being finite Subset of V such that A1: I is Basis of V by Def1; assume dim V = 1 ; ::_thesis: ex v being VECTOR of V st ( v <> 0. V & (Omega). V = Lin {v} ) then card I = 1 by A1, Def2; then consider v being set such that A2: I = {v} by CARD_2:42; v in I by A2, TARSKI:def_1; then reconsider v = v as VECTOR of V ; {v} is linearly-independent by A1, A2, RUSUB_3:def_2; then A3: v <> 0. V by RLVECT_3:8; Lin {v} = UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) by A1, A2, RUSUB_3:def_2; hence ex v being VECTOR of V st ( v <> 0. V & (Omega). V = Lin {v} ) by A3, RUSUB_1:def_3; ::_thesis: verum end; given v being VECTOR of V such that A4: ( v <> 0. V & (Omega). V = Lin {v} ) ; ::_thesis: dim V = 1 ( {v} is linearly-independent & Lin {v} = UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) ) by A4, RLVECT_3:8, RUSUB_1:def_3; then A5: {v} is Basis of V by RUSUB_3:def_2; card {v} = 1 by CARD_1:30; hence dim V = 1 by A5, Def2; ::_thesis: verum end; theorem :: RUSUB_4:14 for V being finite-dimensional RealUnitarySpace holds ( dim V = 2 iff ex u, v being VECTOR of V st ( u <> v & {u,v} is linearly-independent & (Omega). V = Lin {u,v} ) ) proof let V be finite-dimensional RealUnitarySpace; ::_thesis: ( dim V = 2 iff ex u, v being VECTOR of V st ( u <> v & {u,v} is linearly-independent & (Omega). V = Lin {u,v} ) ) hereby ::_thesis: ( ex u, v being VECTOR of V st ( u <> v & {u,v} is linearly-independent & (Omega). V = Lin {u,v} ) implies dim V = 2 ) consider I being finite Subset of V such that A1: I is Basis of V by Def1; assume dim V = 2 ; ::_thesis: ex u, v being VECTOR of V st ( u <> v & {u,v} is linearly-independent & (Omega). V = Lin {u,v} ) then A2: card I = 2 by A1, Def2; then consider u being set such that A3: u in I by CARD_1:27, XBOOLE_0:def_1; reconsider u = u as VECTOR of V by A3; now__::_thesis:_not_I_c=_{u} assume I c= {u} ; ::_thesis: contradiction then card I <= card {u} by NAT_1:43; then 2 <= 1 by A2, CARD_1:30; hence contradiction ; ::_thesis: verum end; then consider v being set such that A4: v in I and A5: not v in {u} by TARSKI:def_3; reconsider v = v as VECTOR of V by A4; A6: v <> u by A5, TARSKI:def_1; A7: now__::_thesis:_I_c=_{u,v} assume not I c= {u,v} ; ::_thesis: contradiction then consider w being set such that A8: w in I and A9: not w in {u,v} by TARSKI:def_3; for x being set st x in {u,v,w} holds x in I by A3, A4, A8, ENUMSET1:def_1; then {u,v,w} c= I by TARSKI:def_3; then A10: card {u,v,w} <= card I by NAT_1:43; ( w <> u & w <> v ) by A9, TARSKI:def_2; then 3 <= 2 by A2, A6, A10, CARD_2:58; hence contradiction ; ::_thesis: verum end; for x being set st x in {u,v} holds x in I by A3, A4, TARSKI:def_2; then {u,v} c= I by TARSKI:def_3; then A11: I = {u,v} by A7, XBOOLE_0:def_10; then A12: {u,v} is linearly-independent by A1, RUSUB_3:def_2; Lin {u,v} = UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) by A1, A11, RUSUB_3:def_2 .= (Omega). V by RUSUB_1:def_3 ; hence ex u, v being VECTOR of V st ( u <> v & {u,v} is linearly-independent & (Omega). V = Lin {u,v} ) by A6, A12; ::_thesis: verum end; given u, v being VECTOR of V such that A13: u <> v and A14: {u,v} is linearly-independent and A15: (Omega). V = Lin {u,v} ; ::_thesis: dim V = 2 Lin {u,v} = UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) by A15, RUSUB_1:def_3; then A16: {u,v} is Basis of V by A14, RUSUB_3:def_2; card {u,v} = 2 by A13, CARD_2:57; hence dim V = 2 by A16, Def2; ::_thesis: verum end; theorem Th15: :: RUSUB_4:15 for V being finite-dimensional RealUnitarySpace for W1, W2 being Subspace of V holds (dim (W1 + W2)) + (dim (W1 /\ W2)) = (dim W1) + (dim W2) proof 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; A4: now__::_thesis:_I1_/\_I2_c=_I 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 set such that A5: x in I1 /\ I2 and A6: not x in I by TARSKI:def_3; 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 by STRUCT_0:def_5; 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_set_st_y_in_I_\/_{x}_holds_ y_in_the_carrier_of_V let y be set ; ::_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 by XBOOLE_1:1; 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; then reconsider Ix = I \/ {x} as Subset of V by TARSKI:def_3; now__::_thesis:_for_y_being_set_st_y_in_I_\/_{x}_holds_ y_in_I19 let y be set ; ::_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; then A10: Ix c= I19 by TARSKI:def_3; 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 by STRUCT_0:def_5; 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) #) by STRUCT_0:def_5; then A12: x in Lin I by A1, RUSUB_3:def_2; Ix \ {x} = I \ {x} by XBOOLE_1:40 .= I by A6, Lm2 ; 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; set A = I1 \/ I2; now__::_thesis:_for_v_being_set_st_v_in_I1_\/_I2_holds_ v_in_the_carrier_of_(W1_+_W2) let v be set ; ::_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, STRUCT_0:def_5; then v9 in W1 + W2 by RUSUB_2:2; hence v in the carrier of (W1 + W2) by STRUCT_0:def_5; ::_thesis: verum end; then reconsider A = I1 \/ I2 as finite Subset of (W1 + W2) by TARSKI:def_3; I c= I1 /\ I2 by A3, A2, XBOOLE_1:19; then I = I1 /\ I2 by A4, XBOOLE_0:def_10; 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 ( 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, XBOOLE_0:def_7 .= {} ; then A30: Carrier L1 misses Carrier L2 by XBOOLE_0:def_7; 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 by STRUCT_0:def_5; 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, STRUCT_0:def_5; 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) assume not Carrier L c= Carrier (L1 + L2) ; ::_thesis: contradiction then consider x being set such that A39: x in Carrier L and A40: not x in Carrier (L1 + L2) by TARSKI:def_3; 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 ; percases ( x in Carrier L1 or x in Carrier L2 ) by A37, A39, XBOOLE_0:def_3; supposeA42: 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; supposeA44: 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; end; end; A46: I \/ I2 = I2 by A2, XBOOLE_1:12; 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, XBOOLE_1:1; 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 by XBOOLE_1:1; 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 A55: A is linearly-independent by RLVECT_3:def_1; 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_set_st_x_in_the_carrier_of_(W1_+_W2)_holds_ x_in_the_carrier_of_(Lin_A9) let x be set ; ::_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 by STRUCT_0:def_5; 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, STRUCT_0:def_5; 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, STRUCT_0:def_5; 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 by XBOOLE_1:1; 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) by STRUCT_0:def_5; ::_thesis: verum end; then the carrier of (W1 + W2) c= the carrier of (Lin A9) by TARSKI:def_3; 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 end; theorem :: RUSUB_4:16 for V being finite-dimensional RealUnitarySpace for W1, W2 being Subspace of V holds dim (W1 /\ W2) >= ((dim W1) + (dim W2)) - (dim V) proof let V be finite-dimensional RealUnitarySpace; ::_thesis: for W1, W2 being Subspace of V holds dim (W1 /\ W2) >= ((dim W1) + (dim W2)) - (dim V) let W1, W2 be Subspace of V; ::_thesis: dim (W1 /\ W2) >= ((dim W1) + (dim W2)) - (dim V) A1: ( dim (W1 + W2) <= dim V & (dim V) + ((dim (W1 /\ W2)) - (dim V)) = dim (W1 /\ W2) ) by Th8; ((dim W1) + (dim W2)) - (dim V) = ((dim (W1 + W2)) + (dim (W1 /\ W2))) - (dim V) by Th15 .= (dim (W1 + W2)) + ((dim (W1 /\ W2)) - (dim V)) ; hence dim (W1 /\ W2) >= ((dim W1) + (dim W2)) - (dim V) by A1, XREAL_1:6; ::_thesis: verum end; theorem :: RUSUB_4:17 for V being finite-dimensional RealUnitarySpace for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds dim V = (dim W1) + (dim W2) proof let V be finite-dimensional RealUnitarySpace; ::_thesis: for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds dim V = (dim W1) + (dim W2) let W1, W2 be Subspace of V; ::_thesis: ( V is_the_direct_sum_of W1,W2 implies dim V = (dim W1) + (dim W2) ) assume A1: V is_the_direct_sum_of W1,W2 ; ::_thesis: dim V = (dim W1) + (dim W2) then A2: UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = W1 + W2 by RUSUB_2:def_4; W1 /\ W2 = (0). V by A1, RUSUB_2:def_4; then (Omega). (W1 /\ W2) = (0). V by RUSUB_1:def_3 .= (0). (W1 /\ W2) by RUSUB_1:30 ; then dim (W1 /\ W2) = 0 by Th12; then (dim W1) + (dim W2) = (dim (W1 + W2)) + 0 by Th15 .= dim ((Omega). V) by A2, RUSUB_1:def_3 .= dim V by Th10 ; hence dim V = (dim W1) + (dim W2) ; ::_thesis: verum end; begin Lm3: for V being finite-dimensional RealUnitarySpace for n being Element of NAT st n <= dim V holds ex W being strict Subspace of V st dim W = n proof let V be finite-dimensional RealUnitarySpace; ::_thesis: for n being Element of NAT st n <= dim V holds ex W being strict Subspace of V st dim W = n let n be Element of NAT ; ::_thesis: ( n <= dim V implies ex W being strict Subspace of V st dim W = n ) consider I being finite Subset of V such that A1: I is Basis of V by Def1; assume n <= dim V ; ::_thesis: ex W being strict Subspace of V st dim W = n then n <= card I by A1, Def2; then consider A being finite Subset of I such that A2: card A = n by FINSEQ_4:72; reconsider A = A as Subset of V by XBOOLE_1:1; reconsider W = Lin A as strict finite-dimensional Subspace of V ; I is linearly-independent by A1, RUSUB_3:def_2; then dim W = n by A2, Th9, RLVECT_3:5; hence ex W being strict Subspace of V st dim W = n ; ::_thesis: verum end; theorem :: RUSUB_4:18 for V being finite-dimensional RealUnitarySpace for W being Subspace of V for n being Element of NAT holds ( n <= dim V iff ex W being strict Subspace of V st dim W = n ) by Lm3, Th8; definition let V be finite-dimensional RealUnitarySpace; let n be Element of NAT ; funcn Subspaces_of V -> set means :Def3: :: RUSUB_4:def 3 for x being set holds ( x in it iff ex W being strict Subspace of V st ( W = x & dim W = n ) ); existence ex b1 being set st for x being set holds ( x in b1 iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) proof set S = { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } ; take { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } ; ::_thesis: for x being set holds ( x in { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) for x being set holds ( x in { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) proof let x be set ; ::_thesis: ( x in { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) hereby ::_thesis: ( ex W being strict Subspace of V st ( W = x & dim W = n ) implies x in { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } ) assume x in { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } ; ::_thesis: ex W being strict Subspace of V st ( W = x & dim W = n ) then A1: ex A being Subset of V st ( x = Lin A & A is linearly-independent & card A = n ) ; then reconsider W = x as strict Subspace of V ; dim W = n by A1, Th9; hence ex W being strict Subspace of V st ( W = x & dim W = n ) ; ::_thesis: verum end; given W being strict Subspace of V such that A2: W = x and A3: dim W = n ; ::_thesis: x in { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } consider A being finite Subset of W such that A4: A is Basis of W by Def1; reconsider A = A as Subset of W ; A is linearly-independent by A4, RUSUB_3:def_2; then reconsider B = A as linearly-independent Subset of V by RUSUB_3:22; A5: x = Lin A by A2, A4, RUSUB_3:def_2 .= Lin B by RUSUB_3:28 ; then card B = n by A2, A3, Th9; hence x in { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } by A5; ::_thesis: verum end; hence for x being set holds ( x in { (Lin A) where A is Subset of V : ( A is linearly-independent & card A = n ) } iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) ; ::_thesis: verum end; uniqueness for b1, b2 being set st ( for x being set holds ( x in b1 iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) ) & ( for x being set holds ( x in b2 iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) ) holds b1 = b2 proof defpred S1[ set ] means ex W being strict Subspace of V st ( W = $1 & dim W = n ); for X1, X2 being set st ( for x being set holds ( x in X1 iff S1[x] ) ) & ( for x being set holds ( x in X2 iff S1[x] ) ) holds X1 = X2 from XBOOLE_0:sch_3(); hence for b1, b2 being set st ( for x being set holds ( x in b1 iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) ) & ( for x being set holds ( x in b2 iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) ) holds b1 = b2 ; ::_thesis: verum end; end; :: deftheorem Def3 defines Subspaces_of RUSUB_4:def_3_:_ for V being finite-dimensional RealUnitarySpace for n being Element of NAT for b3 being set holds ( b3 = n Subspaces_of V iff for x being set holds ( x in b3 iff ex W being strict Subspace of V st ( W = x & dim W = n ) ) ); theorem :: RUSUB_4:19 for V being finite-dimensional RealUnitarySpace for n being Element of NAT st n <= dim V holds not n Subspaces_of V is empty proof let V be finite-dimensional RealUnitarySpace; ::_thesis: for n being Element of NAT st n <= dim V holds not n Subspaces_of V is empty let n be Element of NAT ; ::_thesis: ( n <= dim V implies not n Subspaces_of V is empty ) assume n <= dim V ; ::_thesis: not n Subspaces_of V is empty then ex W being strict Subspace of V st dim W = n by Lm3; hence not n Subspaces_of V is empty by Def3; ::_thesis: verum end; theorem :: RUSUB_4:20 for V being finite-dimensional RealUnitarySpace for n being Element of NAT st dim V < n holds n Subspaces_of V = {} proof let V be finite-dimensional RealUnitarySpace; ::_thesis: for n being Element of NAT st dim V < n holds n Subspaces_of V = {} let n be Element of NAT ; ::_thesis: ( dim V < n implies n Subspaces_of V = {} ) assume that A1: dim V < n and A2: n Subspaces_of V <> {} ; ::_thesis: contradiction consider x being set such that A3: x in n Subspaces_of V by A2, XBOOLE_0:def_1; ex W being strict Subspace of V st ( W = x & dim W = n ) by A3, Def3; hence contradiction by A1, Th8; ::_thesis: verum end; theorem :: RUSUB_4:21 for V being finite-dimensional RealUnitarySpace for W being Subspace of V for n being Element of NAT holds n Subspaces_of W c= n Subspaces_of V proof let V be finite-dimensional RealUnitarySpace; ::_thesis: for W being Subspace of V for n being Element of NAT holds n Subspaces_of W c= n Subspaces_of V let W be Subspace of V; ::_thesis: for n being Element of NAT holds n Subspaces_of W c= n Subspaces_of V let n be Element of NAT ; ::_thesis: n Subspaces_of W c= n Subspaces_of V let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in n Subspaces_of W or x in n Subspaces_of V ) assume x in n Subspaces_of W ; ::_thesis: x in n Subspaces_of V then consider W1 being strict Subspace of W such that A1: W1 = x and A2: dim W1 = n by Def3; reconsider W1 = W1 as strict Subspace of V by RUSUB_1:21; W1 in n Subspaces_of V by A2, Def3; hence x in n Subspaces_of V by A1; ::_thesis: verum end; begin definition let V be non empty RLSStruct ; let S be Subset of V; attrS is Affine means :Def4: :: RUSUB_4:def 4 for x, y being VECTOR of V for a being Real st x in S & y in S holds ((1 - a) * x) + (a * y) in S; end; :: deftheorem Def4 defines Affine RUSUB_4:def_4_:_ for V being non empty RLSStruct for S being Subset of V holds ( S is Affine iff for x, y being VECTOR of V for a being Real st x in S & y in S holds ((1 - a) * x) + (a * y) in S ); theorem Th22: :: RUSUB_4:22 for V being non empty RLSStruct holds ( [#] V is Affine & {} V is Affine ) proof let V be non empty RLSStruct ; ::_thesis: ( [#] V is Affine & {} V is Affine ) for x, y being VECTOR of V for a being Real st x in [#] V & y in [#] V holds ((1 - a) * x) + (a * y) in [#] V ; hence [#] V is Affine by Def4; ::_thesis: {} V is Affine for x, y being VECTOR of V for a being Real st x in {} V & y in {} V holds ((1 - a) * x) + (a * y) in {} V ; hence {} V is Affine by Def4; ::_thesis: verum end; theorem :: RUSUB_4:23 for V being non empty vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct for v being VECTOR of V holds {v} is Affine proof let V be non empty vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct ; ::_thesis: for v being VECTOR of V holds {v} is Affine let v be VECTOR of V; ::_thesis: {v} is Affine for x, y being VECTOR of V for a being Real st x in {v} & y in {v} holds ((1 - a) * x) + (a * y) in {v} proof let x, y be VECTOR of V; ::_thesis: for a being Real st x in {v} & y in {v} holds ((1 - a) * x) + (a * y) in {v} let a be Real; ::_thesis: ( x in {v} & y in {v} implies ((1 - a) * x) + (a * y) in {v} ) assume ( x in {v} & y in {v} ) ; ::_thesis: ((1 - a) * x) + (a * y) in {v} then ( x = v & y = v ) by TARSKI:def_1; then ((1 - a) * x) + (a * y) = ((1 - a) + a) * v by RLVECT_1:def_6 .= v by RLVECT_1:def_8 ; hence ((1 - a) * x) + (a * y) in {v} by TARSKI:def_1; ::_thesis: verum end; hence {v} is Affine by Def4; ::_thesis: verum end; registration let V be non empty RLSStruct ; cluster non empty Affine for Element of K11( the carrier of V); existence ex b1 being Subset of V st ( not b1 is empty & b1 is Affine ) proof take [#] V ; ::_thesis: ( not [#] V is empty & [#] V is Affine ) thus ( not [#] V is empty & [#] V is Affine ) by Th22; ::_thesis: verum end; cluster empty Affine for Element of K11( the carrier of V); existence ex b1 being Subset of V st ( b1 is empty & b1 is Affine ) proof take {} V ; ::_thesis: ( {} V is empty & {} V is Affine ) thus ( {} V is empty & {} V is Affine ) by Th22; ::_thesis: verum end; end; definition let V be RealLinearSpace; let W be Subspace of V; func Up W -> non empty Subset of V equals :: RUSUB_4:def 5 the carrier of W; coherence the carrier of W is non empty Subset of V by RLSUB_1:def_2; end; :: deftheorem defines Up RUSUB_4:def_5_:_ for V being RealLinearSpace for W being Subspace of V holds Up W = the carrier of W; definition let V be RealUnitarySpace; let W be Subspace of V; func Up W -> non empty Subset of V equals :: RUSUB_4:def 6 the carrier of W; coherence the carrier of W is non empty Subset of V by RUSUB_1:def_1; end; :: deftheorem defines Up RUSUB_4:def_6_:_ for V being RealUnitarySpace for W being Subspace of V holds Up W = the carrier of W; theorem :: RUSUB_4:24 for V being RealLinearSpace for W being Subspace of V holds ( Up W is Affine & 0. V in the carrier of W ) proof let V be RealLinearSpace; ::_thesis: for W being Subspace of V holds ( Up W is Affine & 0. V in the carrier of W ) let W be Subspace of V; ::_thesis: ( Up W is Affine & 0. V in the carrier of W ) for x, y being VECTOR of V for a being Real st x in Up W & y in Up W holds ((1 - a) * x) + (a * y) in Up W proof let x, y be VECTOR of V; ::_thesis: for a being Real st x in Up W & y in Up W holds ((1 - a) * x) + (a * y) in Up W let a be Real; ::_thesis: ( x in Up W & y in Up W implies ((1 - a) * x) + (a * y) in Up W ) assume that A1: x in Up W and A2: y in Up W ; ::_thesis: ((1 - a) * x) + (a * y) in Up W y in W by A2, STRUCT_0:def_5; then A3: a * y in W by RLSUB_1:21; x in W by A1, STRUCT_0:def_5; then (1 - a) * x in W by RLSUB_1:21; then ((1 - a) * x) + (a * y) in W by A3, RLSUB_1:20; hence ((1 - a) * x) + (a * y) in Up W by STRUCT_0:def_5; ::_thesis: verum end; hence Up W is Affine by Def4; ::_thesis: 0. V in the carrier of W 0. V in W by RLSUB_1:17; hence 0. V in the carrier of W by STRUCT_0:def_5; ::_thesis: verum end; theorem Th25: :: RUSUB_4:25 for V being RealLinearSpace for A being Affine Subset of V st 0. V in A holds for x being VECTOR of V for a being Real st x in A holds a * x in A proof let V be RealLinearSpace; ::_thesis: for A being Affine Subset of V st 0. V in A holds for x being VECTOR of V for a being Real st x in A holds a * x in A let A be Affine Subset of V; ::_thesis: ( 0. V in A implies for x being VECTOR of V for a being Real st x in A holds a * x in A ) assume A1: 0. V in A ; ::_thesis: for x being VECTOR of V for a being Real st x in A holds a * x in A for x being VECTOR of V for a being Real st x in A holds a * x in A proof let x be VECTOR of V; ::_thesis: for a being Real st x in A holds a * x in A let a be Real; ::_thesis: ( x in A implies a * x in A ) assume x in A ; ::_thesis: a * x in A then ((1 - a) * (0. V)) + (a * x) in A by A1, Def4; then (0. V) + (a * x) in A by RLVECT_1:10; hence a * x in A by RLVECT_1:4; ::_thesis: verum end; hence for x being VECTOR of V for a being Real st x in A holds a * x in A ; ::_thesis: verum end; definition let V be non empty RLSStruct ; let S be non empty Subset of V; attrS is Subspace-like means :Def7: :: RUSUB_4:def 7 ( 0. V in S & ( for x, y being Element of V for a being Real st x in S & y in S holds ( x + y in S & a * x in S ) ) ); end; :: deftheorem Def7 defines Subspace-like RUSUB_4:def_7_:_ for V being non empty RLSStruct for S being non empty Subset of V holds ( S is Subspace-like iff ( 0. V in S & ( for x, y being Element of V for a being Real st x in S & y in S holds ( x + y in S & a * x in S ) ) ) ); theorem Th26: :: RUSUB_4:26 for V being RealLinearSpace for A being non empty Affine Subset of V st 0. V in A holds ( A is Subspace-like & A = the carrier of (Lin A) ) proof let V be RealLinearSpace; ::_thesis: for A being non empty Affine Subset of V st 0. V in A holds ( A is Subspace-like & A = the carrier of (Lin A) ) let A be non empty Affine Subset of V; ::_thesis: ( 0. V in A implies ( A is Subspace-like & A = the carrier of (Lin A) ) ) assume A1: 0. V in A ; ::_thesis: ( A is Subspace-like & A = the carrier of (Lin A) ) A2: for x, y being Element of V for a being Real st x in A & y in A holds ( x + y in A & a * x in A ) proof let x, y be Element of V; ::_thesis: for a being Real st x in A & y in A holds ( x + y in A & a * x in A ) let a be Real; ::_thesis: ( x in A & y in A implies ( x + y in A & a * x in A ) ) assume that A3: x in A and A4: y in A ; ::_thesis: ( x + y in A & a * x in A ) reconsider x = x, y = y as VECTOR of V ; A5: 2 * (((1 - (1 / 2)) * x) + ((1 / 2) * y)) = (2 * ((1 - (1 / 2)) * x)) + (2 * ((1 / 2) * y)) by RLVECT_1:def_5 .= ((2 * (1 - (1 / 2))) * x) + (2 * ((1 / 2) * y)) by RLVECT_1:def_7 .= ((2 - 1) * x) + ((2 * (1 / 2)) * y) by RLVECT_1:def_7 .= x + (1 * y) by RLVECT_1:def_8 .= x + y by RLVECT_1:def_8 ; ((1 - (1 / 2)) * x) + ((1 / 2) * y) in A by A3, A4, Def4; hence ( x + y in A & a * x in A ) by A1, A3, A5, Th25; ::_thesis: verum end; hence A is Subspace-like by A1, Def7; ::_thesis: A = the carrier of (Lin A) for x being set st x in the carrier of (Lin A) holds x in A proof let x be set ; ::_thesis: ( x in the carrier of (Lin A) implies x in A ) assume x in the carrier of (Lin A) ; ::_thesis: x in A then x in Lin A by STRUCT_0:def_5; then A6: ex l being Linear_Combination of A st x = Sum l by RLVECT_3:14; ( ( for v, u being VECTOR of V st v in A & u in A holds v + u in A ) & ( for a being Real for v being VECTOR of V st v in A holds a * v in A ) ) by A2; then A is linearly-closed by RLSUB_1:def_1; hence x in A by A6, RLVECT_2:29; ::_thesis: verum end; then A7: the carrier of (Lin A) c= A by TARSKI:def_3; for x being set st x in A holds x in the carrier of (Lin A) proof let x be set ; ::_thesis: ( x in A implies x in the carrier of (Lin A) ) assume x in A ; ::_thesis: x in the carrier of (Lin A) then x in Lin A by RLVECT_3:15; hence x in the carrier of (Lin A) by STRUCT_0:def_5; ::_thesis: verum end; then A c= the carrier of (Lin A) by TARSKI:def_3; hence A = the carrier of (Lin A) by A7, XBOOLE_0:def_10; ::_thesis: verum end; theorem :: RUSUB_4:27 for V being RealLinearSpace for W being Subspace of V holds Up W is Subspace-like proof let V be RealLinearSpace; ::_thesis: for W being Subspace of V holds Up W is Subspace-like let W be Subspace of V; ::_thesis: Up W is Subspace-like 0. V in W by RLSUB_1:17; hence 0. V in Up W by STRUCT_0:def_5; :: according to RUSUB_4:def_7 ::_thesis: for x, y being Element of V for a being Real st x in Up W & y in Up W holds ( x + y in Up W & a * x in Up W ) thus for x, y being Element of V for a being Real st x in Up W & y in Up W holds ( x + y in Up W & a * x in Up W ) ::_thesis: verum proof let x, y be Element of V; ::_thesis: for a being Real st x in Up W & y in Up W holds ( x + y in Up W & a * x in Up W ) let a be Real; ::_thesis: ( x in Up W & y in Up W implies ( x + y in Up W & a * x in Up W ) ) assume that A1: x in Up W and A2: y in Up W ; ::_thesis: ( x + y in Up W & a * x in Up W ) reconsider x = x, y = y as Element of V ; A3: x in W by A1, STRUCT_0:def_5; then A4: a * x in W by RLSUB_1:21; y in W by A2, STRUCT_0:def_5; then x + y in W by A3, RLSUB_1:20; hence ( x + y in Up W & a * x in Up W ) by A4, STRUCT_0:def_5; ::_thesis: verum end; end; theorem :: RUSUB_4:28 for V being RealUnitarySpace for A being non empty Affine Subset of V st 0. V in A holds A = the carrier of (Lin A) proof let V be RealUnitarySpace; ::_thesis: for A being non empty Affine Subset of V st 0. V in A holds A = the carrier of (Lin A) let A be non empty Affine Subset of V; ::_thesis: ( 0. V in A implies A = the carrier of (Lin A) ) assume 0. V in A ; ::_thesis: A = the carrier of (Lin A) then A1: A is Subspace-like by Th26; for x being set st x in the carrier of (Lin A) holds x in A proof let x be set ; ::_thesis: ( x in the carrier of (Lin A) implies x in A ) assume x in the carrier of (Lin A) ; ::_thesis: x in A then x in Lin A by STRUCT_0:def_5; then A2: ex l being Linear_Combination of A st x = Sum l by RUSUB_3:1; ( ( for v, u being VECTOR of V st v in A & u in A holds v + u in A ) & ( for a being Real for v being VECTOR of V st v in A holds a * v in A ) ) by A1, Def7; then A is linearly-closed by RLSUB_1:def_1; hence x in A by A2, RLVECT_2:29; ::_thesis: verum end; then A3: the carrier of (Lin A) c= A by TARSKI:def_3; for x being set st x in A holds x in the carrier of (Lin A) proof let x be set ; ::_thesis: ( x in A implies x in the carrier of (Lin A) ) assume x in A ; ::_thesis: x in the carrier of (Lin A) then x in Lin A by RUSUB_3:2; hence x in the carrier of (Lin A) by STRUCT_0:def_5; ::_thesis: verum end; then A c= the carrier of (Lin A) by TARSKI:def_3; hence A = the carrier of (Lin A) by A3, XBOOLE_0:def_10; ::_thesis: verum end; theorem :: RUSUB_4:29 for V being RealUnitarySpace for W being Subspace of V holds Up W is Subspace-like proof let V be RealUnitarySpace; ::_thesis: for W being Subspace of V holds Up W is Subspace-like let W be Subspace of V; ::_thesis: Up W is Subspace-like 0. V in W by RUSUB_1:11; hence 0. V in Up W by STRUCT_0:def_5; :: according to RUSUB_4:def_7 ::_thesis: for x, y being Element of V for a being Real st x in Up W & y in Up W holds ( x + y in Up W & a * x in Up W ) thus for x, y being Element of V for a being Real st x in Up W & y in Up W holds ( x + y in Up W & a * x in Up W ) ::_thesis: verum proof let x, y be Element of V; ::_thesis: for a being Real st x in Up W & y in Up W holds ( x + y in Up W & a * x in Up W ) let a be Real; ::_thesis: ( x in Up W & y in Up W implies ( x + y in Up W & a * x in Up W ) ) assume that A1: x in Up W and A2: y in Up W ; ::_thesis: ( x + y in Up W & a * x in Up W ) reconsider x = x, y = y as Element of V ; A3: x in W by A1, STRUCT_0:def_5; then A4: a * x in W by RUSUB_1:15; y in W by A2, STRUCT_0:def_5; then x + y in W by A3, RUSUB_1:14; hence ( x + y in Up W & a * x in Up W ) by A4, STRUCT_0:def_5; ::_thesis: verum end; end; definition let V be non empty addLoopStr ; let M be Subset of V; let v be Element of V; funcv + M -> Subset of V equals :: RUSUB_4:def 8 { (v + u) where u is Element of V : u in M } ; coherence { (v + u) where u is Element of V : u in M } is Subset of V proof set Y = { (v + u) where u is Element of V : u in M } ; defpred S1[ set ] means ex u being Element of V st ( $1 = v + u & u in M ); consider X being set such that A1: for x being set holds ( x in X iff ( x in the carrier of V & S1[x] ) ) from XBOOLE_0:sch_1(); X c= the carrier of V proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in the carrier of V ) assume x in X ; ::_thesis: x in the carrier of V hence x in the carrier of V by A1; ::_thesis: verum end; then reconsider X = X as Subset of V ; reconsider X = X as Subset of V ; A2: { (v + u) where u is Element of V : u in M } c= X proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (v + u) where u is Element of V : u in M } or x in X ) assume x in { (v + u) where u is Element of V : u in M } ; ::_thesis: x in X then ex u being Element of V st ( x = v + u & u in M ) ; hence x in X by A1; ::_thesis: verum end; X c= { (v + u) where u is Element of V : u in M } proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in { (v + u) where u is Element of V : u in M } ) assume x in X ; ::_thesis: x in { (v + u) where u is Element of V : u in M } then ex u being Element of V st ( x = v + u & u in M ) by A1; hence x in { (v + u) where u is Element of V : u in M } ; ::_thesis: verum end; hence { (v + u) where u is Element of V : u in M } is Subset of V by A2, XBOOLE_0:def_10; ::_thesis: verum end; end; :: deftheorem defines + RUSUB_4:def_8_:_ for V being non empty addLoopStr for M being Subset of V for v being Element of V holds v + M = { (v + u) where u is Element of V : u in M } ; theorem :: RUSUB_4:30 for V being RealLinearSpace for W being strict Subspace of V for M being Subset of V for v being VECTOR of V st Up W = M holds v + W = v + M proof let V be RealLinearSpace; ::_thesis: for W being strict Subspace of V for M being Subset of V for v being VECTOR of V st Up W = M holds v + W = v + M let W be strict Subspace of V; ::_thesis: for M being Subset of V for v being VECTOR of V st Up W = M holds v + W = v + M let M be Subset of V; ::_thesis: for v being VECTOR of V st Up W = M holds v + W = v + M let v be VECTOR of V; ::_thesis: ( Up W = M implies v + W = v + M ) assume A1: Up W = M ; ::_thesis: v + W = v + M for x being set st x in v + M holds x in v + W proof let x be set ; ::_thesis: ( x in v + M implies x in v + W ) assume x in v + M ; ::_thesis: x in v + W then consider u being Element of V such that A2: x = v + u and A3: u in M ; u in W by A1, A3, STRUCT_0:def_5; then x in { (v + u9) where u9 is VECTOR of V : u9 in W } by A2; hence x in v + W by RLSUB_1:def_5; ::_thesis: verum end; then A4: v + M c= v + W by TARSKI:def_3; for x being set st x in v + W holds x in v + M proof let x be set ; ::_thesis: ( x in v + W implies x in v + M ) assume x in v + W ; ::_thesis: x in v + M then x in { (v + u) where u is VECTOR of V : u in W } by RLSUB_1:def_5; then consider u being VECTOR of V such that A5: x = v + u and A6: u in W ; u in M by A1, A6, STRUCT_0:def_5; hence x in v + M by A5; ::_thesis: verum end; then v + W c= v + M by TARSKI:def_3; hence v + W = v + M by A4, XBOOLE_0:def_10; ::_thesis: verum end; theorem Th31: :: RUSUB_4:31 for V being non empty Abelian add-associative vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct for M being Affine Subset of V for v being VECTOR of V holds v + M is Affine proof let V be non empty Abelian add-associative vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct ; ::_thesis: for M being Affine Subset of V for v being VECTOR of V holds v + M is Affine let M be Affine Subset of V; ::_thesis: for v being VECTOR of V holds v + M is Affine let v be VECTOR of V; ::_thesis: v + M is Affine for x, y being VECTOR of V for a being Real st x in v + M & y in v + M holds ((1 - a) * x) + (a * y) in v + M proof let x, y be VECTOR of V; ::_thesis: for a being Real st x in v + M & y in v + M holds ((1 - a) * x) + (a * y) in v + M let a be Real; ::_thesis: ( x in v + M & y in v + M implies ((1 - a) * x) + (a * y) in v + M ) assume that A1: x in v + M and A2: y in v + M ; ::_thesis: ((1 - a) * x) + (a * y) in v + M consider x9 being Element of V such that A3: x = v + x9 and A4: x9 in M by A1; consider y9 being Element of V such that A5: y = v + y9 and A6: y9 in M by A2; A7: ((1 - a) * x) + (a * y) = (((1 - a) * v) + ((1 - a) * x9)) + (a * (v + y9)) by A3, A5, RLVECT_1:def_5 .= (((1 - a) * v) + ((1 - a) * x9)) + ((a * v) + (a * y9)) by RLVECT_1:def_5 .= ((((1 - a) * v) + ((1 - a) * x9)) + (a * v)) + (a * y9) by RLVECT_1:def_3 .= (((1 - a) * x9) + (((1 - a) * v) + (a * v))) + (a * y9) by RLVECT_1:def_3 .= (((1 - a) * x9) + (((1 - a) + a) * v)) + (a * y9) by RLVECT_1:def_6 .= (((1 - a) * x9) + v) + (a * y9) by RLVECT_1:def_8 .= v + (((1 - a) * x9) + (a * y9)) by RLVECT_1:def_3 ; ((1 - a) * x9) + (a * y9) in M by A4, A6, Def4; hence ((1 - a) * x) + (a * y) in v + M by A7; ::_thesis: verum end; hence v + M is Affine by Def4; ::_thesis: verum end; theorem :: RUSUB_4:32 for V being RealUnitarySpace for W being strict Subspace of V for M being Subset of V for v being VECTOR of V st Up W = M holds v + W = v + M proof let V be RealUnitarySpace; ::_thesis: for W being strict Subspace of V for M being Subset of V for v being VECTOR of V st Up W = M holds v + W = v + M let W be strict Subspace of V; ::_thesis: for M being Subset of V for v being VECTOR of V st Up W = M holds v + W = v + M let M be Subset of V; ::_thesis: for v being VECTOR of V st Up W = M holds v + W = v + M let v be VECTOR of V; ::_thesis: ( Up W = M implies v + W = v + M ) assume A1: Up W = M ; ::_thesis: v + W = v + M for x being set st x in v + M holds x in v + W proof let x be set ; ::_thesis: ( x in v + M implies x in v + W ) assume x in v + M ; ::_thesis: x in v + W then consider u being Element of V such that A2: x = v + u and A3: u in M ; u in W by A1, A3, STRUCT_0:def_5; then x in { (v + u9) where u9 is VECTOR of V : u9 in W } by A2; hence x in v + W by RUSUB_1:def_4; ::_thesis: verum end; then A4: v + M c= v + W by TARSKI:def_3; for x being set st x in v + W holds x in v + M proof let x be set ; ::_thesis: ( x in v + W implies x in v + M ) assume x in v + W ; ::_thesis: x in v + M then x in { (v + u) where u is VECTOR of V : u in W } by RUSUB_1:def_4; then consider u being VECTOR of V such that A5: x = v + u and A6: u in W ; u in M by A1, A6, STRUCT_0:def_5; hence x in v + M by A5; ::_thesis: verum end; then v + W c= v + M by TARSKI:def_3; hence v + W = v + M by A4, XBOOLE_0:def_10; ::_thesis: verum end; definition let V be non empty addLoopStr ; let M, N be Subset of V; funcM + N -> Subset of V equals :: RUSUB_4:def 9 { (u + v) where u, v is Element of V : ( u in M & v in N ) } ; coherence { (u + v) where u, v is Element of V : ( u in M & v in N ) } is Subset of V proof defpred S1[ set , set ] means ( $1 in M & $2 in N ); deffunc H1( Element of V, Element of V) -> Element of the carrier of V = $1 + $2; { H1(u,v) where u, v is Element of V : S1[u,v] } is Subset of V from DOMAIN_1:sch_9(); hence { (u + v) where u, v is Element of V : ( u in M & v in N ) } is Subset of V ; ::_thesis: verum end; end; :: deftheorem defines + RUSUB_4:def_9_:_ for V being non empty addLoopStr for M, N being Subset of V holds M + N = { (u + v) where u, v is Element of V : ( u in M & v in N ) } ; definition let V be non empty Abelian addLoopStr ; let M, N be Subset of V; :: original: + redefine funcM + N -> Subset of V; commutativity for M, N being Subset of V holds M + N = N + M proof let M, N be Subset of V; ::_thesis: M + N = N + M for x being set st x in M + N holds x in N + M proof let x be set ; ::_thesis: ( x in M + N implies x in N + M ) assume x in M + N ; ::_thesis: x in N + M then ex u1, v1 being Element of V st ( x = u1 + v1 & u1 in M & v1 in N ) ; hence x in N + M ; ::_thesis: verum end; then A1: M + N c= N + M by TARSKI:def_3; for x being set st x in N + M holds x in M + N proof let x be set ; ::_thesis: ( x in N + M implies x in M + N ) assume x in N + M ; ::_thesis: x in M + N then ex u1, v1 being Element of V st ( x = u1 + v1 & u1 in N & v1 in M ) ; hence x in M + N ; ::_thesis: verum end; then N + M c= M + N by TARSKI:def_3; hence M + N = N + M by A1, XBOOLE_0:def_10; ::_thesis: verum end; end; theorem Th33: :: RUSUB_4:33 for V being non empty addLoopStr for M being Subset of V for v being Element of V holds {v} + M = v + M proof let V be non empty addLoopStr ; ::_thesis: for M being Subset of V for v being Element of V holds {v} + M = v + M let M be Subset of V; ::_thesis: for v being Element of V holds {v} + M = v + M let v be Element of V; ::_thesis: {v} + M = v + M for x being set st x in v + M holds x in {v} + M proof let x be set ; ::_thesis: ( x in v + M implies x in {v} + M ) assume x in v + M ; ::_thesis: x in {v} + M then A1: ex u being Element of V st ( x = v + u & u in M ) ; v in {v} by TARSKI:def_1; hence x in {v} + M by A1; ::_thesis: verum end; then A2: v + M c= {v} + M by TARSKI:def_3; for x being set st x in {v} + M holds x in v + M proof let x be set ; ::_thesis: ( x in {v} + M implies x in v + M ) assume x in {v} + M ; ::_thesis: x in v + M then consider v1, u1 being Element of V such that A3: x = v1 + u1 and A4: v1 in {v} and A5: u1 in M ; v1 = v by A4, TARSKI:def_1; hence x in v + M by A3, A5; ::_thesis: verum end; then {v} + M c= v + M by TARSKI:def_3; hence {v} + M = v + M by A2, XBOOLE_0:def_10; ::_thesis: verum end; theorem :: RUSUB_4:34 for V being non empty Abelian add-associative vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct for M being Affine Subset of V for v being VECTOR of V holds {v} + M is Affine proof let V be non empty Abelian add-associative vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct ; ::_thesis: for M being Affine Subset of V for v being VECTOR of V holds {v} + M is Affine let M be Affine Subset of V; ::_thesis: for v being VECTOR of V holds {v} + M is Affine let v be VECTOR of V; ::_thesis: {v} + M is Affine {v} + M = v + M by Th33; hence {v} + M is Affine by Th31; ::_thesis: verum end; theorem :: RUSUB_4:35 for V being non empty RLSStruct for M, N being Affine Subset of V holds M /\ N is Affine proof let V be non empty RLSStruct ; ::_thesis: for M, N being Affine Subset of V holds M /\ N is Affine let M, N be Affine Subset of V; ::_thesis: M /\ N is Affine for x, y being VECTOR of V for a being Real st x in M /\ N & y in M /\ N holds ((1 - a) * x) + (a * y) in M /\ N proof let x, y be VECTOR of V; ::_thesis: for a being Real st x in M /\ N & y in M /\ N holds ((1 - a) * x) + (a * y) in M /\ N let a be Real; ::_thesis: ( x in M /\ N & y in M /\ N implies ((1 - a) * x) + (a * y) in M /\ N ) assume A1: ( x in M /\ N & y in M /\ N ) ; ::_thesis: ((1 - a) * x) + (a * y) in M /\ N then ( x in N & y in N ) by XBOOLE_0:def_4; then A2: ((1 - a) * x) + (a * y) in N by Def4; ( x in M & y in M ) by A1, XBOOLE_0:def_4; then ((1 - a) * x) + (a * y) in M by Def4; hence ((1 - a) * x) + (a * y) in M /\ N by A2, XBOOLE_0:def_4; ::_thesis: verum end; hence M /\ N is Affine by Def4; ::_thesis: verum end; theorem :: RUSUB_4:36 for V being non empty Abelian add-associative vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct for M, N being Affine Subset of V holds M + N is Affine proof let V be non empty Abelian add-associative vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct ; ::_thesis: for M, N being Affine Subset of V holds M + N is Affine let M, N be Affine Subset of V; ::_thesis: M + N is Affine for x, y being VECTOR of V for a being Real st x in M + N & y in M + N holds ((1 - a) * x) + (a * y) in M + N proof let x, y be VECTOR of V; ::_thesis: for a being Real st x in M + N & y in M + N holds ((1 - a) * x) + (a * y) in M + N let a be Real; ::_thesis: ( x in M + N & y in M + N implies ((1 - a) * x) + (a * y) in M + N ) assume that A1: x in M + N and A2: y in M + N ; ::_thesis: ((1 - a) * x) + (a * y) in M + N consider u1, v1 being Element of V such that A3: x = u1 + v1 and A4: ( u1 in M & v1 in N ) by A1; consider u2, v2 being Element of V such that A5: y = u2 + v2 and A6: ( u2 in M & v2 in N ) by A2; A7: ((1 - a) * x) + (a * y) = (((1 - a) * u1) + ((1 - a) * v1)) + (a * (u2 + v2)) by A3, A5, RLVECT_1:def_5 .= (((1 - a) * u1) + ((1 - a) * v1)) + ((a * u2) + (a * v2)) by RLVECT_1:def_5 .= ((((1 - a) * u1) + ((1 - a) * v1)) + (a * u2)) + (a * v2) by RLVECT_1:def_3 .= (((1 - a) * v1) + (((1 - a) * u1) + (a * u2))) + (a * v2) by RLVECT_1:def_3 .= (((1 - a) * u1) + (a * u2)) + (((1 - a) * v1) + (a * v2)) by RLVECT_1:def_3 ; ( ((1 - a) * u1) + (a * u2) in M & ((1 - a) * v1) + (a * v2) in N ) by A4, A6, Def4; hence ((1 - a) * x) + (a * y) in M + N by A7; ::_thesis: verum end; hence M + N is Affine by Def4; ::_thesis: verum end;