let n be Nat; :: thesis:
let x0 be Element of (RealVectSpace (Seg n)); :: thesis:
let B be Subset of (RealVectSpace (Seg n)); :: thesis:
reconsider x1 = x0 as Element of REAL n by FINSEQ_2:93;
A1: REAL n = the carrier of (RealVectSpace (Seg n)) by FINSEQ_2:93;
assume A2: B = RN_Base n ; :: thesis:
A3: { x where x is Element of REAL n : ex i being Element of NAT st
( 1 <= i & i <= n & x = Base_FinSeq (n,i) ) } c= B

let y be set ; :: according to TARSKI:def 3 :: thesis:
assume y in { x where x is Element of REAL n : ex i being Element of NAT st
( 1 <= i & i <= n & x = Base_FinSeq (n,i) ) } ; :: thesis:
then ex x being Element of REAL n st
( y = x & ex i being Element of NAT st
( 1 <= i & i <= n & x = Base_FinSeq (n,i) ) ) ;
hence y in B by A2; :: thesis:

end;

B c= { x where x is Element of REAL n : ex i being Element of NAT st
( 1 <= i & i <= n & x = Base_FinSeq (n,i) ) }

proof

let y be set ; :: according to TARSKI:def 3 :: thesis:
assume y in B ; :: thesis:
then ex i2 being Element of NAT st
( y = Base_FinSeq (n,i2) & 1 <= i2 & i2 <= n ) by A2;
hence y in { x where x is Element of REAL n : ex i being Element of NAT st
( 1 <= i & i <= n & x = Base_FinSeq (n,i) ) } ; :: thesis:

end;

then A4: B = { x where x is Element of REAL n : ex i being Element of NAT st
( 1 <= i & i <= n & x = Base_FinSeq (n,i) ) } by A3, XBOOLE_0:def 10;
A5: { x where x is Element of REAL n : ex i being Element of NAT st
( 1 <= i & i <= n & x = Base_FinSeq (n,i) & |(x1,x)| <> 0 ) } c= B

proof

let x2 be set ; :: according to TARSKI:def 3 :: thesis:
assume x2 in { x where x is Element of REAL n : ex i being Element of NAT st
( 1 <= i & i <= n & x = Base_FinSeq (n,i) & |(x1,x)| <> 0 ) } ; :: thesis:
then ex x being Element of REAL n st
( x = x2 & ex i being Element of NAT st
( 1 <= i & i <= n & x = Base_FinSeq (n,i) & |(x1,x)| <> 0 ) ) ;
hence x2 in B by A4; :: thesis:

end;

then reconsider B0 = { x where x is Element of REAL n : ex i being Element of NAT st
( 1 <= i & i <= n & x = Base_FinSeq (n,i) & |(x1,x)| <> 0 ) } as Subset of (RealVectSpace (Seg n)) by XBOOLE_1:1;
A6: dom (ProjFinSeq x1) = Seg (len (ProjFinSeq x1)) by FINSEQ_1:def 3
.= Seg n by Def12 ;
defpred S₁[ set , set ] means ( $1 in B0 implies ex i being Element of NAT st
( $2 = i & 1 <= i & i <= n & $1 = Base_FinSeq (n,i) ) );
A7: rng (ProjFinSeq x1) c= REAL n ;
A8: for x being set st x in B0 holds
ex y being set st
( y in Seg n & S₁[x,y] )

proof

let x be set ; :: thesis:
assume x in B0 ; :: thesis:
then consider x2 being Element of REAL n such that
A9: x = x2 and
A10: ex i being Element of NAT st
( 1 <= i & i <= n & x2 = Base_FinSeq (n,i) & |(x1,x2)| <> 0 ) ;
consider i0 being Element of NAT such that
A11: 1 <= i0 and
A12: i0 <= n and
A13: x2 = Base_FinSeq (n,i0) and
|(x1,x2)| <> 0 by A10;
i0 in Seg n by A11, A12, FINSEQ_1:1;
hence ex y being set st
( y in Seg n & S₁[x,y] ) by A9, A11, A12, A13; :: thesis:

end;

consider f being Function of B0,(Seg n) such that
A14: for x being set st x in B0 holds
S₁[x,f . x] from FUNCT_2:sch 1(A8);
defpred S₂[ set , set ] means ( ( $1 in B0 implies for i being Element of NAT st 1 <= i & i <= n & $1 = Base_FinSeq (n,i) holds
$2 = |(x1,(Base_FinSeq (n,i)))| ) & ( not $1 in B0 implies $2 = 0 ) );
A15: for x being set st x in the carrier of (RealVectSpace (Seg n)) holds
ex y being set st
( y in REAL & S₂[x,y] )

proof

let x be set ; :: thesis:
assume x in the carrier of (RealVectSpace (Seg n)) ; :: thesis:

per cases ;

suppose A16: x in B0 ; :: thesis:

then consider x2 being Element of REAL n such that
A17: x2 = x and
ex i being Element of NAT st
( 1 <= i & i <= n & x2 = Base_FinSeq (n,i) & |(x1,x2)| <> 0 ) ;
reconsider y0 = |(x1,x2)| as Element of REAL ;
for i2 being Element of NAT st 1 <= i2 & i2 <= n & x = Base_FinSeq (n,i2) holds
y0 = |(x1,(Base_FinSeq (n,i2)))| by A17;
hence ex y being set st
( y in REAL & S₂[x,y] ) by A16; :: thesis:

end;

suppose not x in B0 ; :: thesis:

hence ex y being set st
( y in REAL & S₂[x,y] ) ; :: thesis:

end;

consider l2 being Function of the carrier of (RealVectSpace (Seg n)),REAL such that
A18: for x being set st x in the carrier of (RealVectSpace (Seg n)) holds
S₂[x,l2 . x] from FUNCT_2:sch 1(A15);
A19: l2 is Element of Funcs ( the carrier of (RealVectSpace (Seg n)),REAL) by FUNCT_2:8;
for v being Element of (RealVectSpace (Seg n)) st not v in B0 holds
l2 . v = 0 by A18;
then reconsider l3 = l2 as Linear_Combination of RealVectSpace (Seg n) by A2, A5, A19, RLVECT_2:def 3;
Carrier l3 c= B0

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in Carrier l3 ; :: thesis:
then x in { v where v is Element of (RealVectSpace (Seg n)) : l3 . v <> 0 } by RLVECT_2:def 4;
then ex v being Element of (RealVectSpace (Seg n)) st
( x = v & l3 . v <> 0 ) ;
hence x in B0 by A18; :: thesis:

end;

then reconsider l0 = l3 as Linear_Combination of B0 by RLVECT_2:def 6;
A20: Carrier l0 c= B0 by RLVECT_2:def 6;
then Carrier l0 c= B by A5, XBOOLE_1:1;
then reconsider l2 = l0 as Linear_Combination of B by RLVECT_2:def 6;
A21: B0 c= Carrier l0

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume A22: x in B0 ; :: thesis:
then consider x6 being Element of REAL n such that
A23: x = x6 and
A24: ex i being Element of NAT st
( 1 <= i & i <= n & x6 = Base_FinSeq (n,i) & |(x1,x6)| <> 0 ) ;
reconsider x66 = x6 as Element of (RealVectSpace (Seg n)) by FINSEQ_2:93;
consider i8 being Element of NAT such that
1 <= i8 and
i8 <= n and
A25: x6 = Base_FinSeq (n,i8) and
|(x1,x6)| <> 0 by A24;
l0 . x66 = |(x1,(Base_FinSeq (n,i8)))| by A18, A22, A23, A24, A25;
then x in { v where v is Element of (RealVectSpace (Seg n)) : l0 . v <> 0 } by A23, A24, A25;
hence x in Carrier l0 by RLVECT_2:def 4; :: thesis:

end;

for x1, x2 being set st x1 in dom f & x2 in dom f & f . x1 = f . x2 holds
x1 = x2

proof

let x1, x2 be set ; :: thesis:
assume that
A26: x1 in dom f and
A27: x2 in dom f and
A28: f . x1 = f . x2 ; :: thesis:
A29: ex i2 being Element of NAT st
( f . x2 = i2 & 1 <= i2 & i2 <= n & x2 = Base_FinSeq (n,i2) ) by A14, A27;
ex i1 being Element of NAT st
( f . x1 = i1 & 1 <= i1 & i1 <= n & x1 = Base_FinSeq (n,i1) ) by A14, A26;
hence x1 = x2 by A28, A29; :: thesis:

end;

then A30: f is one-to-one by FUNCT_1:def 4;
A31: ( Seg n = {} implies B0 = {} )

proof

assume A32: Seg n = {} ; :: thesis:

now

set y = the Element of B0;
assume B0 <> {} ; :: thesis:
then the Element of B0 in B0 ;
then ex x being Element of REAL n st
( x = the Element of B0 & ex i being Element of NAT st
( 1 <= i & i <= n & x = Base_FinSeq (n,i) & |(x1,x)| <> 0 ) ) ;
hence contradiction by A32; :: thesis:

end;

hence B0 = {} ; :: thesis:

end;

A33: for i3 being Element of NAT st i3 in dom (ProjFinSeq x1) & not i3 in rng (Sgm (rng f)) holds
(ProjFinSeq x1) . i3 = 0* n

proof

let i3 be Element of NAT ; :: thesis:
assume that
A34: i3 in dom (ProjFinSeq x1) and
A35: not i3 in rng (Sgm (rng f)) ; :: thesis:
A36: i3 in Seg (len (ProjFinSeq x1)) by A34, FINSEQ_1:def 3;
then A37: 1 <= i3 by FINSEQ_1:1;
len (ProjFinSeq x1) = n by Def12;
then A38: i3 <= n by A36, FINSEQ_1:1;
A39: not i3 in rng f by A35, FINSEQ_1:def 13;

A40: now

assume |(x1,(Base_FinSeq (n,i3)))| <> 0 ; :: thesis:
then A41: Base_FinSeq (n,i3) in B0 by A37, A38;
then consider i5 being Element of NAT such that
A42: f . (Base_FinSeq (n,i3)) = i5 and
1 <= i5 and
i5 <= n and
A43: Base_FinSeq (n,i3) = Base_FinSeq (n,i5) by A14;
A44: Base_FinSeq (n,i3) in dom f by A31, A41, FUNCT_2:def 1;
i3 = i5 by A37, A38, A43, Th27;
hence contradiction by A39, A42, A44, FUNCT_1:def 3; :: thesis:

end;

(ProjFinSeq x1) . i3 = |(x1,(Base_FinSeq (n,i3)))| * (Base_FinSeq (n,i3)) by A37, A38, Def12;
hence (ProjFinSeq x1) . i3 = 0* n by A40, EUCLID_4:3; :: thesis:

end;

A45: dom (Sgm (rng f)) = Seg (len (Sgm (rng f))) by FINSEQ_1:def 3;
rng ((ProjFinSeq x1) * (Sgm (rng f))) c= rng (ProjFinSeq x1) by RELAT_1:26;
then A46: rng ((ProjFinSeq x1) * (Sgm (rng f))) c= REAL n by A7, XBOOLE_1:1;
A47: rng (Sgm (rng f)) = rng f by FINSEQ_1:def 13;
dom ((ProjFinSeq x1) * (Sgm (rng f))) = (Sgm (rng f)) " (dom (ProjFinSeq x1)) by RELAT_1:147
.= dom (Sgm (rng f)) by A47, A6, Th2 ;
then (ProjFinSeq x1) * (Sgm (rng f)) is FinSequence by A45, FINSEQ_1:def 2;
then reconsider F2 = (ProjFinSeq x1) * (Sgm (rng f)) as FinSequence of the carrier of (RealVectSpace (Seg n)) by A1, A46, FINSEQ_1:def 4;
dom F2 = (Sgm (rng f)) " (Seg n) by A6, RELAT_1:147
.= dom (Sgm (rng f)) by A47, Th2 ;
then A48: dom F2 = Seg (len (Sgm (rng f))) by FINSEQ_1:def 3;
then A49: Seg (len F2) = Seg (len (Sgm (rng f))) by FINSEQ_1:def 3;
reconsider F3 = F2 as FinSequence of REAL n by FINSEQ_2:93;
A50: x0 = Sum (ProjFinSeq x1) by Th33
.= Sum F3 by A47, A6, A33, Th26, FINSEQ_3:92
.= Sum F2 by Th45 ;
A51: rng ((f ") * (Sgm (rng f))) c= rng (f ") by RELAT_1:26;
A52: dom (Sgm (rng f)) = Seg (len (Sgm (rng f))) by FINSEQ_1:def 3;
A53: len F2 = len (Sgm (rng f)) by A48, FINSEQ_1:def 3;
A54: dom f = B0 by A31, FUNCT_2:def 1;
then rng (f ") = B0 by A30, FUNCT_1:33;
then A55: rng ((f ") * (Sgm (rng f))) c= the carrier of (RealVectSpace (Seg n)) by A51, XBOOLE_1:1;
dom ((f ") * (Sgm (rng f))) = (Sgm (rng f)) " (dom (f ")) by RELAT_1:147
.= (Sgm (rng f)) " (rng f) by A30, FUNCT_1:33
.= dom (Sgm (rng f)) by A47, Th2 ;
then (f ") * (Sgm (rng f)) is FinSequence by A52, FINSEQ_1:def 2;
then reconsider F0 = (f ") * (Sgm (rng f)) as FinSequence of the carrier of (RealVectSpace (Seg n)) by A55, FINSEQ_1:def 4;
dom F0 = (Sgm (rng f)) " (dom (f ")) by RELAT_1:147
.= (Sgm (rng f)) " (rng f) by A30, FUNCT_1:33
.= dom (Sgm (rng f)) by A47, Th2 ;
then A56: dom F0 = Seg (len (Sgm (rng f))) by FINSEQ_1:def 3;
dom (f ") = rng f by A30, FUNCT_1:33;
then rng F0 = rng (f ") by A47, RELAT_1:28
.= dom f by A30, FUNCT_1:33 ;
then A57: rng F0 = Carrier l0 by A54, A20, A21, XBOOLE_0:def 10;
A58: for i being Element of NAT st i in dom F2 holds
F2 . i = (l0 . (F0 /. i)) * (F0 /. i)

proof

end;

A78: Sgm (rng f) is one-to-one by FINSEQ_3:92;
len F2 = len F0 by A56, A49, FINSEQ_1:def 3;
then x1 = Sum (l0 (#) F0) by A50, A58, RLVECT_2:def 7;
then x1 = Sum l2 by A30, A78, A57, RLVECT_2:def 8;
hence ex l being Linear_Combination of B st x0 = Sum l ; :: thesis: