let m be Nat; :: thesis:
let K be Field; :: thesis:
let W be strict Subspace of m -VectSp_over K; :: thesis:
assume A1: dim W < m ; :: thesis:

per cases ;

suppose A2: dim W = 0 ; :: thesis:

then m -' (dim W) = m by NAT_D:40;
then reconsider ONE = 1. (K,m) as Matrix of m -' (dim W),m,K ;
take ONE ; :: thesis:
take Seg m ; :: thesis:
A3: len (1. (K,m)) = m by MATRIX_0:24;
A4: dim (Space_of_Solutions_of ONE) = 0 by Lm8;
A5: m -' (dim W) = m by A2, NAT_D:40;
A6: width (1. (K,m)) = m by MATRIX_0:24;
Space_of_Solutions_of ONE = (Omega). (Space_of_Solutions_of ONE)
.= (0). (Space_of_Solutions_of ONE) by A4, VECTSP_9:29
.= (0). W by A6, VECTSP_4:37
.= (Omega). W by A2, VECTSP_9:29
.= W ;
hence ( card (Seg m) = m -' (dim W) & Seg m c= Seg m & Segm (ONE,(Seg (m -' (dim W))),(Seg m)) = 1. (K,(m -' (dim W))) & W = Space_of_Solutions_of ONE ) by A5, A3, A6, FINSEQ_1:57, MATRIX13:46; :: thesis:

end;

suppose A7: dim W > 0 ; :: thesis:

set ZERO = 0. (K,(m -' (dim W)),m);
A8: m - (dim W) > (dim W) - (dim W) by A1, XREAL_1:9;
A9: m -' (dim W) = m - (dim W) by A1, XREAL_1:233;
then A10: ( len (0. (K,(m -' (dim W)),m)) = m -' (dim W) & width (0. (K,(m -' (dim W)),m)) = m ) by A8, MATRIX_0:23;
A11: card (Seg m) = m by FINSEQ_1:57;
consider A being Matrix of dim W,m,K, N being finite without_zero Subset of NAT such that
A12: N c= Seg m and
A13: dim W = card N and
A14: Segm (A,(Seg (dim W)),N) = 1. (K,(dim W)) and
the_rank_of A = dim W and
A15: lines A is Basis of W by Th70;
set SA = Segm (A,(Seg (dim W)),((Seg m) \ N));
A16: card ((Seg m) \ N) = (card (Seg m)) - (card N) by A12, CARD_2:44;
then A17: width (Segm (A,(Seg (dim W)),((Seg m) \ N))) = m - (card N) by A7, A11, MATRIX_0:23;
A18: card (Seg (dim W)) = dim W by FINSEQ_1:57;
then len (Segm (A,(Seg (dim W)),((Seg m) \ N))) = dim W by A7, MATRIX_0:23;
then width ((Segm (A,(Seg (dim W)),((Seg m) \ N))) @) = dim W by A13, A8, A17, MATRIX_0:54;
then A19: width (- ((Segm (A,(Seg (dim W)),((Seg m) \ N))) @)) = dim W by MATRIX_3:def 2;
A20: card (Seg (m -' (dim W))) = m -' (dim W) by FINSEQ_1:57;
m -' (dim W) = m - (dim W) by A1, XREAL_1:233;
then reconsider CC = 1. (K,(m -' (dim W))) as Matrix of card (Seg (m -' (dim W))), card ((Seg m) \ N),K by A13, A16, A11, A20;
A21: ( (Seg m) \ ((Seg m) \ N) = (Seg m) /\ N & m -' (m -' (dim W)) = m - (m -' (dim W)) ) by NAT_D:35, XBOOLE_1:48, XREAL_1:233;
A22: Indices (0. (K,(m -' (dim W)),m)) = [:(Seg (m -' (dim W))),(Seg m):] by A9, A8, MATRIX_0:23;
then A23: [:(Seg (m -' (dim W))),N:] c= Indices (0. (K,(m -' (dim W)),m)) by A12, ZFMISC_1:95;
len ((Segm (A,(Seg (dim W)),((Seg m) \ N))) @) = m - (dim W) by A13, A8, A17, MATRIX_0:54;
then len (- ((Segm (A,(Seg (dim W)),((Seg m) \ N))) @)) = m -' (dim W) by A9, MATRIX_3:def 2;
then reconsider BB = - ((Segm (A,(Seg (dim W)),((Seg m) \ N))) @) as Matrix of card (Seg (m -' (dim W))), card N,K by A13, A20, A19, MATRIX_0:51;
A24: N misses (Seg m) \ N by XBOOLE_1:79;
A25: (Seg m) \ N c= Seg m by XBOOLE_1:36;
then A26: [:(Seg (m -' (dim W))),((Seg m) \ N):] c= Indices (0. (K,(m -' (dim W)),m)) by A22, ZFMISC_1:95;
[:(Seg (m -' (dim W))),N:] /\ [:(Seg (m -' (dim W))),((Seg m) \ N):] = [:(Seg (m -' (dim W))),(N /\ ((Seg m) \ N)):] by ZFMISC_1:99
.= [:(Seg (m -' (dim W))),{}:] by A24
.= {} by ZFMISC_1:90 ;
then for i, j, bi, bj, ci, cj being Nat st [i,j] in [:(Seg (m -' (dim W))),N:] /\ [:(Seg (m -' (dim W))),((Seg m) \ N):] & bi = ((Sgm (Seg (m -' (dim W)))) ") . i & bj = ((Sgm N) ") . j & ci = ((Sgm (Seg (m -' (dim W)))) ") . i & cj = ((Sgm ((Seg m) \ N)) ") . j holds
BB * (bi,bj) = CC * (ci,cj) ;
then consider M being Matrix of m -' (dim W),m,K such that
A27: Segm (M,(Seg (m -' (dim W))),N) = BB and
A28: Segm (M,(Seg (m -' (dim W))),((Seg m) \ N)) = CC and
for i, j being Nat st [i,j] in (Indices M) \ ([:(Seg (m -' (dim W))),N:] \/ [:(Seg (m -' (dim W))),((Seg m) \ N):]) holds
M * (i,j) = (0. (K,(m -' (dim W)),m)) * (i,j) by A10, A23, A26, Th9;
(Seg m) /\ N = N by A12, XBOOLE_1:28;
then consider MV being Matrix of dim W,m,K such that
A29: Segm (MV,(Seg (dim W)),N) = 1. (K,(dim W)) and
A30: Segm (MV,(Seg (dim W)),((Seg m) \ N)) = - ((Segm (M,(Seg (m -' (dim W))),N)) @) and
A31: Lin (lines MV) = Space_of_Solutions_of M by A7, A13, A9, A8, A16, A11, A28, A21, Th67, XBOOLE_1:36;

A32: now :: thesis:

A33: Indices A = [:(Seg (dim W)),(Seg m):] by A7, MATRIX_0:23;
let i, j be Nat; :: thesis:
assume A34: [i,j] in Indices A ; :: thesis:
A35: i in Seg (dim W) by A34, A33, ZFMISC_1:87;
A36: Indices A = Indices MV by MATRIX_0:26;
A37: rng (Sgm (Seg (dim W))) = Seg (dim W) by FINSEQ_1:def 14;
dom (Sgm (Seg (dim W))) = Seg (dim W) by A18, FINSEQ_3:40;
then consider x being object such that
A38: x in Seg (dim W) and
A39: (Sgm (Seg (dim W))) . x = i by A35, A37, FUNCT_1:def 3;
reconsider x = x as Element of NAT by A38;
A40: j in Seg m by A34, A33, ZFMISC_1:87;

now :: thesis:

per cases ;

suppose A41: j in N ; :: thesis:

then A42: [i,j] in [:(Seg (dim W)),N:] by A35, ZFMISC_1:87;
A43: rng (Sgm N) = N by FINSEQ_1:def 14;
dom (Sgm N) = Seg (dim W) by A13, FINSEQ_3:40;
then consider y being object such that
A44: y in Seg (dim W) and
A45: (Sgm N) . y = j by A41, A43, FUNCT_1:def 3;
reconsider y = y as Element of NAT by A44;
A46: [:(Seg (dim W)),N:] c= Indices A by A12, A33, ZFMISC_1:95;
then A47: [x,y] in Indices (Segm (MV,(Seg (dim W)),N)) by A36, A37, A39, A43, A45, A42, MATRIX13:17;
[x,y] in Indices (Segm (A,(Seg (dim W)),N)) by A37, A39, A43, A45, A46, A42, MATRIX13:17;
hence A * (i,j) = (Segm (MV,(Seg (dim W)),N)) * (x,y) by A14, A29, A39, A45, MATRIX13:def 1
.= MV * (i,j) by A39, A45, A47, MATRIX13:def 1 ;
:: thesis:

end;

suppose not j in N ; :: thesis:

then A48: j in (Seg m) \ N by A40, XBOOLE_0:def 5;
then A49: [i,j] in [:(Seg (dim W)),((Seg m) \ N):] by A35, ZFMISC_1:87;
A50: rng (Sgm ((Seg m) \ N)) = (Seg m) \ N by FINSEQ_1:def 14;
dom (Sgm ((Seg m) \ N)) = Seg (m -' (dim W)) by A13, A9, A16, A11, FINSEQ_3:40;
then consider y being object such that
A51: y in Seg (m -' (dim W)) and
A52: (Sgm ((Seg m) \ N)) . y = j by A48, A50, FUNCT_1:def 3;
reconsider y = y as Element of NAT by A51;
A53: [:(Seg (dim W)),((Seg m) \ N):] c= Indices A by A25, A33, ZFMISC_1:95;
then A54: [x,y] in Indices (Segm (A,(Seg (dim W)),((Seg m) \ N))) by A37, A39, A50, A52, A49, MATRIX13:17;
A55: [x,y] in Indices (Segm (MV,(Seg (dim W)),((Seg m) \ N))) by A36, A37, A39, A50, A52, A53, A49, MATRIX13:17;
then A56: [x,y] in Indices ((Segm (M,(Seg (m -' (dim W))),N)) @) by A30, Lm1;
then A57: [y,x] in Indices (Segm (M,(Seg (m -' (dim W))),N)) by MATRIX_0:def 6;
then A58: [y,x] in Indices ((Segm (A,(Seg (dim W)),((Seg m) \ N))) @) by A27, Lm1;
thus MV * (i,j) = (- ((Segm (M,(Seg (m -' (dim W))),N)) @)) * (x,y) by A30, A39, A52, A55, MATRIX13:def 1
.= - (((Segm (M,(Seg (m -' (dim W))),N)) @) * (x,y)) by A56, MATRIX_3:def 2
.= - ((- ((Segm (A,(Seg (dim W)),((Seg m) \ N))) @)) * (y,x)) by A27, A57, MATRIX_0:def 6
.= - (- (((Segm (A,(Seg (dim W)),((Seg m) \ N))) @) * (y,x))) by A58, MATRIX_3:def 2
.= ((Segm (A,(Seg (dim W)),((Seg m) \ N))) @) * (y,x) by RLVECT_1:17
.= (Segm (A,(Seg (dim W)),((Seg m) \ N))) * (x,y) by A54, MATRIX_0:def 6
.= A * (i,j) by A39, A52, A54, MATRIX13:def 1 ; :: thesis:

end;

hence A * (i,j) = MV * (i,j) ; :: thesis:

end;

then reconsider lA = lines MV as Subset of W by A15, MATRIX_0:27;
take M ; :: thesis:
take (Seg m) \ N ; :: thesis:
MV = A by A32, MATRIX_0:27;
then Lin lA = ModuleStr(# the carrier of W, the addF of W, the ZeroF of W, the lmult of W #) by A15, VECTSP_7:def 3;
hence ( card ((Seg m) \ N) = m -' (dim W) & (Seg m) \ N c= Seg m & Segm (M,(Seg (m -' (dim W))),((Seg m) \ N)) = 1. (K,(m -' (dim W))) & W = Space_of_Solutions_of M ) by A1, A13, A16, A11, A28, A31, VECTSP_9:17, XBOOLE_1:36, XREAL_1:233; :: thesis:

end;