let V be RealLinearSpace; :: thesis: for A being Subset of V holds

( ( A <> {} & A is linearly-closed ) iff for l being Linear_Combination of A holds Sum l in A )

let A be Subset of V; :: thesis: ( ( A <> {} & A is linearly-closed ) iff for l being Linear_Combination of A holds Sum l in A )

thus ( A <> {} & A is linearly-closed implies for l being Linear_Combination of A holds Sum l in A ) :: thesis: ( ( for l being Linear_Combination of A holds Sum l in A ) implies ( A <> {} & A is linearly-closed ) )

hence A <> {} ; :: thesis: A is linearly-closed

( ZeroLC V is Linear_Combination of A & Sum (ZeroLC V) = 0. V ) by Lm2, Th22;

then A52: 0. V in A by A51;

A53: for a being Real

for v being VECTOR of V st v in A holds

a * v in A

v + u in A :: according to RLSUB_1:def 1 :: thesis: for b_{1} being object

for b_{2} being Element of the carrier of V holds

( not b_{2} in A or b_{1} * b_{2} in A )_{1} being object

for b_{2} being Element of the carrier of V holds

( not b_{2} in A or b_{1} * b_{2} in A )
by A53; :: thesis: verum

( ( A <> {} & A is linearly-closed ) iff for l being Linear_Combination of A holds Sum l in A )

let A be Subset of V; :: thesis: ( ( A <> {} & A is linearly-closed ) iff for l being Linear_Combination of A holds Sum l in A )

thus ( A <> {} & A is linearly-closed implies for l being Linear_Combination of A holds Sum l in A ) :: thesis: ( ( for l being Linear_Combination of A holds Sum l in A ) implies ( A <> {} & A is linearly-closed ) )

proof

assume A51:
for l being Linear_Combination of A holds Sum l in A
; :: thesis: ( A <> {} & A is linearly-closed )
defpred S_{1}[ Nat] means for l being Linear_Combination of A st card (Carrier l) = $1 holds

Sum l in A;

assume that

A1: A <> {} and

A2: A is linearly-closed ; :: thesis: for l being Linear_Combination of A holds Sum l in A

_{1}[ 0 ]
;

_{1}[k] holds

S_{1}[k + 1]
;

let l be Linear_Combination of A; :: thesis: Sum l in A

A50: card (Carrier l) = card (Carrier l) ;

for k being Nat holds S_{1}[k]
from NAT_1:sch 2(A3, A49);

hence Sum l in A by A50; :: thesis: verum

end;Sum l in A;

assume that

A1: A <> {} and

A2: A is linearly-closed ; :: thesis: for l being Linear_Combination of A holds Sum l in A

now :: thesis: for l being Linear_Combination of A st card (Carrier l) = 0 holds

Sum l in A

then A3:
SSum l in A

let l be Linear_Combination of A; :: thesis: ( card (Carrier l) = 0 implies Sum l in A )

assume card (Carrier l) = 0 ; :: thesis: Sum l in A

then Carrier l = {} ;

then l = ZeroLC V by Def5;

then Sum l = 0. V by Lm2;

hence Sum l in A by A1, A2, RLSUB_1:1; :: thesis: verum

end;assume card (Carrier l) = 0 ; :: thesis: Sum l in A

then Carrier l = {} ;

then l = ZeroLC V by Def5;

then Sum l = 0. V by Lm2;

hence Sum l in A by A1, A2, RLSUB_1:1; :: thesis: verum

now :: thesis: for k being Nat st ( for l being Linear_Combination of A st card (Carrier l) = k holds

Sum l in A ) holds

for l being Linear_Combination of A st card (Carrier l) = k + 1 holds

Sum l in A

then A49:
for k being Nat st SSum l in A ) holds

for l being Linear_Combination of A st card (Carrier l) = k + 1 holds

Sum l in A

let k be Nat; :: thesis: ( ( for l being Linear_Combination of A st card (Carrier l) = k holds

Sum l in A ) implies for l being Linear_Combination of A st card (Carrier l) = k + 1 holds

Sum l in A )

assume A4: for l being Linear_Combination of A st card (Carrier l) = k holds

Sum l in A ; :: thesis: for l being Linear_Combination of A st card (Carrier l) = k + 1 holds

Sum l in A

let l be Linear_Combination of A; :: thesis: ( card (Carrier l) = k + 1 implies Sum l in A )

deffunc H_{1}( Element of V) -> Element of REAL = l . $1;

consider F being FinSequence of V such that

A5: F is one-to-one and

A6: rng F = Carrier l and

A7: Sum l = Sum (l (#) F) by Def8;

reconsider G = F | (Seg k) as FinSequence of the carrier of V by FINSEQ_1:18;

assume A8: card (Carrier l) = k + 1 ; :: thesis: Sum l in A

then A9: len F = k + 1 by A5, A6, FINSEQ_4:62;

then A10: len (l (#) F) = k + 1 by Def7;

A11: k + 1 in Seg (k + 1) by FINSEQ_1:4;

then A12: k + 1 in dom F by A9, FINSEQ_1:def 3;

k + 1 in dom F by A9, A11, FINSEQ_1:def 3;

then reconsider v = F . (k + 1) as VECTOR of V by FUNCT_1:102;

consider f being Function of the carrier of V,REAL such that

A13: f . v = In (0,REAL) and

A14: for u being Element of V st u <> v holds

f . u = H_{1}(u)
from FUNCT_2:sch 6();

reconsider f = f as Element of Funcs ( the carrier of V,REAL) by FUNCT_2:8;

A15: v in Carrier l by A6, A12, FUNCT_1:def 3;

A17: A \ {v} c= A by XBOOLE_1:36;

A18: Carrier l c= A by Def6;

then A19: (l . v) * v in A by A2, A15;

A20: Carrier f = (Carrier l) \ {v}

then Carrier f c= A by A17;

then reconsider f = f as Linear_Combination of A by Def6;

A27: len G = k by A9, FINSEQ_3:53;

then A28: len (f (#) G) = k by Def7;

A29: rng G = Carrier f

.= dom (f (#) G) by A28, FINSEQ_1:def 3 ;

then A39: dom (f (#) G) = (dom (l (#) F)) /\ (Seg k) by A10, FINSEQ_1:def 3;

v in rng F by A12, FUNCT_1:def 3;

then {v} c= Carrier l by A6, ZFMISC_1:31;

then card (Carrier f) = (k + 1) - (card {v}) by A8, A20, CARD_2:44

.= (k + 1) - 1 by CARD_1:30

.= k ;

then A47: Sum f in A by A4;

G is one-to-one by A5, FUNCT_1:52;

then A48: Sum (f (#) G) = Sum f by A29, Def8;

( dom (f (#) G) = Seg (len (f (#) G)) & (l (#) F) . (len F) = (l . v) * v ) by A9, A12, Th24, FINSEQ_1:def 3;

then Sum (l (#) F) = (Sum (f (#) G)) + ((l . v) * v) by A9, A10, A28, A46, RLVECT_1:38;

hence Sum l in A by A2, A7, A19, A48, A47; :: thesis: verum

end;Sum l in A ) implies for l being Linear_Combination of A st card (Carrier l) = k + 1 holds

Sum l in A )

assume A4: for l being Linear_Combination of A st card (Carrier l) = k holds

Sum l in A ; :: thesis: for l being Linear_Combination of A st card (Carrier l) = k + 1 holds

Sum l in A

let l be Linear_Combination of A; :: thesis: ( card (Carrier l) = k + 1 implies Sum l in A )

deffunc H

consider F being FinSequence of V such that

A5: F is one-to-one and

A6: rng F = Carrier l and

A7: Sum l = Sum (l (#) F) by Def8;

reconsider G = F | (Seg k) as FinSequence of the carrier of V by FINSEQ_1:18;

assume A8: card (Carrier l) = k + 1 ; :: thesis: Sum l in A

then A9: len F = k + 1 by A5, A6, FINSEQ_4:62;

then A10: len (l (#) F) = k + 1 by Def7;

A11: k + 1 in Seg (k + 1) by FINSEQ_1:4;

then A12: k + 1 in dom F by A9, FINSEQ_1:def 3;

k + 1 in dom F by A9, A11, FINSEQ_1:def 3;

then reconsider v = F . (k + 1) as VECTOR of V by FUNCT_1:102;

consider f being Function of the carrier of V,REAL such that

A13: f . v = In (0,REAL) and

A14: for u being Element of V st u <> v holds

f . u = H

reconsider f = f as Element of Funcs ( the carrier of V,REAL) by FUNCT_2:8;

A15: v in Carrier l by A6, A12, FUNCT_1:def 3;

now :: thesis: for u being VECTOR of V st not u in Carrier l holds

f . u = 0

then reconsider f = f as Linear_Combination of V by Def3;f . u = 0

let u be VECTOR of V; :: thesis: ( not u in Carrier l implies f . u = 0 )

assume A16: not u in Carrier l ; :: thesis: f . u = 0

hence f . u = l . u by A15, A14

.= 0 by A16 ;

:: thesis: verum

end;assume A16: not u in Carrier l ; :: thesis: f . u = 0

hence f . u = l . u by A15, A14

.= 0 by A16 ;

:: thesis: verum

A17: A \ {v} c= A by XBOOLE_1:36;

A18: Carrier l c= A by Def6;

then A19: (l . v) * v in A by A2, A15;

A20: Carrier f = (Carrier l) \ {v}

proof

then
Carrier f c= A \ {v}
by A18, XBOOLE_1:33;
thus
Carrier f c= (Carrier l) \ {v}
:: according to XBOOLE_0:def 10 :: thesis: (Carrier l) \ {v} c= Carrier f

assume A24: x in (Carrier l) \ {v} ; :: thesis: x in Carrier f

then x in Carrier l by XBOOLE_0:def 5;

then consider u being VECTOR of V such that

A25: x = u and

A26: l . u <> 0 ;

not x in {v} by A24, XBOOLE_0:def 5;

then x <> v by TARSKI:def 1;

then l . u = f . u by A14, A25;

hence x in Carrier f by A25, A26; :: thesis: verum

end;proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (Carrier l) \ {v} or x in Carrier f )
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Carrier f or x in (Carrier l) \ {v} )

assume x in Carrier f ; :: thesis: x in (Carrier l) \ {v}

then consider u being VECTOR of V such that

A21: u = x and

A22: f . u <> 0 ;

f . u = l . u by A13, A14, A22;

then A23: x in Carrier l by A21, A22;

not x in {v} by A13, A21, A22, TARSKI:def 1;

hence x in (Carrier l) \ {v} by A23, XBOOLE_0:def 5; :: thesis: verum

end;assume x in Carrier f ; :: thesis: x in (Carrier l) \ {v}

then consider u being VECTOR of V such that

A21: u = x and

A22: f . u <> 0 ;

f . u = l . u by A13, A14, A22;

then A23: x in Carrier l by A21, A22;

not x in {v} by A13, A21, A22, TARSKI:def 1;

hence x in (Carrier l) \ {v} by A23, XBOOLE_0:def 5; :: thesis: verum

assume A24: x in (Carrier l) \ {v} ; :: thesis: x in Carrier f

then x in Carrier l by XBOOLE_0:def 5;

then consider u being VECTOR of V such that

A25: x = u and

A26: l . u <> 0 ;

not x in {v} by A24, XBOOLE_0:def 5;

then x <> v by TARSKI:def 1;

then l . u = f . u by A14, A25;

hence x in Carrier f by A25, A26; :: thesis: verum

then Carrier f c= A by A17;

then reconsider f = f as Linear_Combination of A by Def6;

A27: len G = k by A9, FINSEQ_3:53;

then A28: len (f (#) G) = k by Def7;

A29: rng G = Carrier f

proof

(Seg (k + 1)) /\ (Seg k) =
Seg k
by FINSEQ_1:7, NAT_1:12
thus
rng G c= Carrier f
:: according to XBOOLE_0:def 10 :: thesis: Carrier f c= rng G

assume A35: x in Carrier f ; :: thesis: x in rng G

then x in rng F by A6, A20, XBOOLE_0:def 5;

then consider y being object such that

A36: y in dom F and

A37: F . y = x by FUNCT_1:def 3;

reconsider y = y as Element of NAT by A36;

then A38: y in dom G by RELAT_1:61;

then G . y = F . y by FUNCT_1:47;

hence x in rng G by A37, A38, FUNCT_1:def 3; :: thesis: verum

end;proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Carrier f or x in rng G )
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng G or x in Carrier f )

assume x in rng G ; :: thesis: x in Carrier f

then consider y being object such that

A30: y in dom G and

A31: G . y = x by FUNCT_1:def 3;

reconsider y = y as Element of NAT by A30;

A32: ( dom G c= dom F & G . y = F . y ) by A30, FUNCT_1:47, RELAT_1:60;

x in rng F by A30, A31, A32, FUNCT_1:def 3;

hence x in Carrier f by A6, A20, A34, XBOOLE_0:def 5; :: thesis: verum

end;assume x in rng G ; :: thesis: x in Carrier f

then consider y being object such that

A30: y in dom G and

A31: G . y = x by FUNCT_1:def 3;

reconsider y = y as Element of NAT by A30;

A32: ( dom G c= dom F & G . y = F . y ) by A30, FUNCT_1:47, RELAT_1:60;

now :: thesis: not x = v

then A34:
not x in {v}
by TARSKI:def 1;assume
x = v
; :: thesis: contradiction

then A33: k + 1 = y by A5, A12, A30, A31, A32;

y <= k by A27, A30, FINSEQ_3:25;

hence contradiction by A33, XREAL_1:29; :: thesis: verum

end;then A33: k + 1 = y by A5, A12, A30, A31, A32;

y <= k by A27, A30, FINSEQ_3:25;

hence contradiction by A33, XREAL_1:29; :: thesis: verum

x in rng F by A30, A31, A32, FUNCT_1:def 3;

hence x in Carrier f by A6, A20, A34, XBOOLE_0:def 5; :: thesis: verum

assume A35: x in Carrier f ; :: thesis: x in rng G

then x in rng F by A6, A20, XBOOLE_0:def 5;

then consider y being object such that

A36: y in dom F and

A37: F . y = x by FUNCT_1:def 3;

reconsider y = y as Element of NAT by A36;

now :: thesis: y in Seg k

then
y in (dom F) /\ (Seg k)
by A36, XBOOLE_0:def 4;assume
not y in Seg k
; :: thesis: contradiction

then y in (dom F) \ (Seg k) by A36, XBOOLE_0:def 5;

then y in (Seg (k + 1)) \ (Seg k) by A9, FINSEQ_1:def 3;

then y in {(k + 1)} by FINSEQ_3:15;

then y = k + 1 by TARSKI:def 1;

then not v in {v} by A20, A35, A37, XBOOLE_0:def 5;

hence contradiction by TARSKI:def 1; :: thesis: verum

end;then y in (dom F) \ (Seg k) by A36, XBOOLE_0:def 5;

then y in (Seg (k + 1)) \ (Seg k) by A9, FINSEQ_1:def 3;

then y in {(k + 1)} by FINSEQ_3:15;

then y = k + 1 by TARSKI:def 1;

then not v in {v} by A20, A35, A37, XBOOLE_0:def 5;

hence contradiction by TARSKI:def 1; :: thesis: verum

then A38: y in dom G by RELAT_1:61;

then G . y = F . y by FUNCT_1:47;

hence x in rng G by A37, A38, FUNCT_1:def 3; :: thesis: verum

.= dom (f (#) G) by A28, FINSEQ_1:def 3 ;

then A39: dom (f (#) G) = (dom (l (#) F)) /\ (Seg k) by A10, FINSEQ_1:def 3;

now :: thesis: for x being object st x in dom (f (#) G) holds

(f (#) G) . x = (l (#) F) . x

then A46:
f (#) G = (l (#) F) | (Seg k)
by A39, FUNCT_1:46;(f (#) G) . x = (l (#) F) . x

let x be object ; :: thesis: ( x in dom (f (#) G) implies (f (#) G) . x = (l (#) F) . x )

assume A40: x in dom (f (#) G) ; :: thesis: (f (#) G) . x = (l (#) F) . x

then reconsider n = x as Element of NAT ;

n in dom (l (#) F) by A39, A40, XBOOLE_0:def 4;

then A41: n in dom F by A9, A10, FINSEQ_3:29;

then F . n in rng F by FUNCT_1:def 3;

then reconsider w = F . n as VECTOR of V ;

A42: n in dom G by A27, A28, A40, FINSEQ_3:29;

then A43: G . n in rng G by FUNCT_1:def 3;

then reconsider u = G . n as VECTOR of V ;

not u in {v} by A20, A29, A43, XBOOLE_0:def 5;

then A44: u <> v by TARSKI:def 1;

A45: (f (#) G) . n = (f . u) * u by A42, Th24

.= (l . u) * u by A14, A44 ;

w = u by A42, FUNCT_1:47;

hence (f (#) G) . x = (l (#) F) . x by A45, A41, Th24; :: thesis: verum

end;assume A40: x in dom (f (#) G) ; :: thesis: (f (#) G) . x = (l (#) F) . x

then reconsider n = x as Element of NAT ;

n in dom (l (#) F) by A39, A40, XBOOLE_0:def 4;

then A41: n in dom F by A9, A10, FINSEQ_3:29;

then F . n in rng F by FUNCT_1:def 3;

then reconsider w = F . n as VECTOR of V ;

A42: n in dom G by A27, A28, A40, FINSEQ_3:29;

then A43: G . n in rng G by FUNCT_1:def 3;

then reconsider u = G . n as VECTOR of V ;

not u in {v} by A20, A29, A43, XBOOLE_0:def 5;

then A44: u <> v by TARSKI:def 1;

A45: (f (#) G) . n = (f . u) * u by A42, Th24

.= (l . u) * u by A14, A44 ;

w = u by A42, FUNCT_1:47;

hence (f (#) G) . x = (l (#) F) . x by A45, A41, Th24; :: thesis: verum

v in rng F by A12, FUNCT_1:def 3;

then {v} c= Carrier l by A6, ZFMISC_1:31;

then card (Carrier f) = (k + 1) - (card {v}) by A8, A20, CARD_2:44

.= (k + 1) - 1 by CARD_1:30

.= k ;

then A47: Sum f in A by A4;

G is one-to-one by A5, FUNCT_1:52;

then A48: Sum (f (#) G) = Sum f by A29, Def8;

( dom (f (#) G) = Seg (len (f (#) G)) & (l (#) F) . (len F) = (l . v) * v ) by A9, A12, Th24, FINSEQ_1:def 3;

then Sum (l (#) F) = (Sum (f (#) G)) + ((l . v) * v) by A9, A10, A28, A46, RLVECT_1:38;

hence Sum l in A by A2, A7, A19, A48, A47; :: thesis: verum

S

let l be Linear_Combination of A; :: thesis: Sum l in A

A50: card (Carrier l) = card (Carrier l) ;

for k being Nat holds S

hence Sum l in A by A50; :: thesis: verum

hence A <> {} ; :: thesis: A is linearly-closed

( ZeroLC V is Linear_Combination of A & Sum (ZeroLC V) = 0. V ) by Lm2, Th22;

then A52: 0. V in A by A51;

A53: for a being Real

for v being VECTOR of V st v in A holds

a * v in A

proof

thus
for v, u being VECTOR of V st v in A & u in A holds
let a be Real; :: thesis: for v being VECTOR of V st v in A holds

a * v in A

let v be VECTOR of V; :: thesis: ( v in A implies a * v in A )

assume A54: v in A ; :: thesis: a * v in A

end;a * v in A

let v be VECTOR of V; :: thesis: ( v in A implies a * v in A )

assume A54: v in A ; :: thesis: a * v in A

now :: thesis: a * v in Aend;

hence
a * v in A
; :: thesis: verumper cases
( a = 0 or a <> 0 )
;

end;

suppose A55:
a <> 0
; :: thesis: a * v in A

deffunc H_{1}( Element of V) -> Element of REAL = zz;

reconsider aa = a as Element of REAL by XREAL_0:def 1;

consider f being Function of the carrier of V,REAL such that

A56: f . v = aa and

A57: for u being Element of V st u <> v holds

f . u = H_{1}(u)
from FUNCT_2:sch 6();

reconsider f = f as Element of Funcs ( the carrier of V,REAL) by FUNCT_2:8;

A58: Carrier f = {v}

then reconsider f = f as Linear_Combination of A by A58, Def6;

consider F being FinSequence of V such that

A61: ( F is one-to-one & rng F = Carrier f ) and

A62: Sum (f (#) F) = Sum f by Def8;

F = <*v*> by A58, A61, FINSEQ_3:97;

then f (#) F = <*((f . v) * v)*> by Th26;

then Sum f = a * v by A56, A62, RLVECT_1:44;

hence a * v in A by A51; :: thesis: verum

end;reconsider aa = a as Element of REAL by XREAL_0:def 1;

consider f being Function of the carrier of V,REAL such that

A56: f . v = aa and

A57: for u being Element of V st u <> v holds

f . u = H

reconsider f = f as Element of Funcs ( the carrier of V,REAL) by FUNCT_2:8;

now :: thesis: for u being VECTOR of V st not u in {v} holds

f . u = 0

then reconsider f = f as Linear_Combination of V by Def3;f . u = 0

let u be VECTOR of V; :: thesis: ( not u in {v} implies f . u = 0 )

assume not u in {v} ; :: thesis: f . u = 0

then u <> v by TARSKI:def 1;

hence f . u = 0 by A57; :: thesis: verum

end;assume not u in {v} ; :: thesis: f . u = 0

then u <> v by TARSKI:def 1;

hence f . u = 0 by A57; :: thesis: verum

A58: Carrier f = {v}

proof

{v} c= A
by A54, ZFMISC_1:31;
thus
Carrier f c= {v}
:: according to XBOOLE_0:def 10 :: thesis: {v} c= Carrier f

assume x in {v} ; :: thesis: x in Carrier f

then x = v by TARSKI:def 1;

hence x in Carrier f by A55, A56; :: thesis: verum

end;proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {v} or x in Carrier f )
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Carrier f or x in {v} )

assume x in Carrier f ; :: thesis: x in {v}

then consider u being VECTOR of V such that

A59: x = u and

A60: f . u <> 0 ;

u = v by A57, A60;

hence x in {v} by A59, TARSKI:def 1; :: thesis: verum

end;assume x in Carrier f ; :: thesis: x in {v}

then consider u being VECTOR of V such that

A59: x = u and

A60: f . u <> 0 ;

u = v by A57, A60;

hence x in {v} by A59, TARSKI:def 1; :: thesis: verum

assume x in {v} ; :: thesis: x in Carrier f

then x = v by TARSKI:def 1;

hence x in Carrier f by A55, A56; :: thesis: verum

then reconsider f = f as Linear_Combination of A by A58, Def6;

consider F being FinSequence of V such that

A61: ( F is one-to-one & rng F = Carrier f ) and

A62: Sum (f (#) F) = Sum f by Def8;

F = <*v*> by A58, A61, FINSEQ_3:97;

then f (#) F = <*((f . v) * v)*> by Th26;

then Sum f = a * v by A56, A62, RLVECT_1:44;

hence a * v in A by A51; :: thesis: verum

v + u in A :: according to RLSUB_1:def 1 :: thesis: for b

for b

( not b

proof

thus
for b
let v, u be VECTOR of V; :: thesis: ( v in A & u in A implies v + u in A )

assume that

A63: v in A and

A64: u in A ; :: thesis: v + u in A

end;assume that

A63: v in A and

A64: u in A ; :: thesis: v + u in A

now :: thesis: v + u in Aend;

hence
v + u in A
; :: thesis: verumper cases
( u = v or v <> u )
;

end;

suppose
u = v
; :: thesis: v + u in A

then v + u =
(1 * v) + v
by RLVECT_1:def 8

.= (1 * v) + (1 * v) by RLVECT_1:def 8

.= (1 + 1) * v by RLVECT_1:def 6

.= 2 * v ;

hence v + u in A by A53, A63; :: thesis: verum

end;.= (1 * v) + (1 * v) by RLVECT_1:def 8

.= (1 + 1) * v by RLVECT_1:def 6

.= 2 * v ;

hence v + u in A by A53, A63; :: thesis: verum

suppose A65:
v <> u
; :: thesis: v + u in A

deffunc H_{1}( Element of V) -> Element of REAL = zz;

reconsider jj = 1 as Element of REAL by XREAL_0:def 1;

consider f being Function of the carrier of V,REAL such that

A66: ( f . v = jj & f . u = jj ) and

A67: for w being Element of V st w <> v & w <> u holds

f . w = H_{1}(w)
from FUNCT_2:sch 7(A65);

reconsider f = f as Element of Funcs ( the carrier of V,REAL) by FUNCT_2:8;

A68: Carrier f = {v,u}

A70: ( 1 * u = u & 1 * v = v ) by RLVECT_1:def 8;

reconsider f = f as Linear_Combination of A by A69, Def6;

consider F being FinSequence of V such that

A71: ( F is one-to-one & rng F = Carrier f ) and

A72: Sum (f (#) F) = Sum f by Def8;

( F = <*v,u*> or F = <*u,v*> ) by A65, A68, A71, FINSEQ_3:99;

then ( f (#) F = <*(1 * v),(1 * u)*> or f (#) F = <*(1 * u),(1 * v)*> ) by A66, Th27;

then Sum f = v + u by A72, A70, RLVECT_1:45;

hence v + u in A by A51; :: thesis: verum

end;reconsider jj = 1 as Element of REAL by XREAL_0:def 1;

consider f being Function of the carrier of V,REAL such that

A66: ( f . v = jj & f . u = jj ) and

A67: for w being Element of V st w <> v & w <> u holds

f . w = H

reconsider f = f as Element of Funcs ( the carrier of V,REAL) by FUNCT_2:8;

now :: thesis: for w being VECTOR of V st not w in {v,u} holds

f . w = 0

then reconsider f = f as Linear_Combination of V by Def3;f . w = 0

let w be VECTOR of V; :: thesis: ( not w in {v,u} implies f . w = 0 )

assume not w in {v,u} ; :: thesis: f . w = 0

then ( w <> v & w <> u ) by TARSKI:def 2;

hence f . w = 0 by A67; :: thesis: verum

end;assume not w in {v,u} ; :: thesis: f . w = 0

then ( w <> v & w <> u ) by TARSKI:def 2;

hence f . w = 0 by A67; :: thesis: verum

A68: Carrier f = {v,u}

proof

then A69:
Carrier f c= A
by A63, A64, ZFMISC_1:32;
thus
Carrier f c= {v,u}
:: according to XBOOLE_0:def 10 :: thesis: {v,u} c= Carrier f

assume x in {v,u} ; :: thesis: x in Carrier f

then ( x = v or x = u ) by TARSKI:def 2;

hence x in Carrier f by A66; :: thesis: verum

end;proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {v,u} or x in Carrier f )
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Carrier f or x in {v,u} )

assume x in Carrier f ; :: thesis: x in {v,u}

then ex w being VECTOR of V st

( x = w & f . w <> 0 ) ;

then ( x = v or x = u ) by A67;

hence x in {v,u} by TARSKI:def 2; :: thesis: verum

end;assume x in Carrier f ; :: thesis: x in {v,u}

then ex w being VECTOR of V st

( x = w & f . w <> 0 ) ;

then ( x = v or x = u ) by A67;

hence x in {v,u} by TARSKI:def 2; :: thesis: verum

assume x in {v,u} ; :: thesis: x in Carrier f

then ( x = v or x = u ) by TARSKI:def 2;

hence x in Carrier f by A66; :: thesis: verum

A70: ( 1 * u = u & 1 * v = v ) by RLVECT_1:def 8;

reconsider f = f as Linear_Combination of A by A69, Def6;

consider F being FinSequence of V such that

A71: ( F is one-to-one & rng F = Carrier f ) and

A72: Sum (f (#) F) = Sum f by Def8;

( F = <*v,u*> or F = <*u,v*> ) by A65, A68, A71, FINSEQ_3:99;

then ( f (#) F = <*(1 * v),(1 * u)*> or f (#) F = <*(1 * u),(1 * v)*> ) by A66, Th27;

then Sum f = v + u by A72, A70, RLVECT_1:45;

hence v + u in A by A51; :: thesis: verum

for b

( not b