let K be Field; :: thesis: for A being Matrix of K
for N being finite without_zero Subset of NAT st N c= dom A & not N is empty & width A > 0 & ( for i being Nat st i in (dom A) \ N holds
Line (A,i) = (width A) |-> (0. K) ) holds
Space_of_Solutions_of A = Space_of_Solutions_of (Segm (A,N,(Seg (width A))))
let A be Matrix of K; :: thesis: for N being finite without_zero Subset of NAT st N c= dom A & not N is empty & width A > 0 & ( for i being Nat st i in (dom A) \ N holds
Line (A,i) = (width A) |-> (0. K) ) holds
Space_of_Solutions_of A = Space_of_Solutions_of (Segm (A,N,(Seg (width A))))
let N be finite without_zero Subset of NAT; :: thesis: ( N c= dom A & not N is empty & width A > 0 & ( for i being Nat st i in (dom A) \ N holds
Line (A,i) = (width A) |-> (0. K) ) implies Space_of_Solutions_of A = Space_of_Solutions_of (Segm (A,N,(Seg (width A)))) )
assume that
A1: N c= dom A and
A2: not N is empty and
A3: width A > 0 and
A4: for i being Nat st i in (dom A) \ N holds
Line (A,i) = (width A) |-> (0. K) ; :: thesis: Space_of_Solutions_of A = Space_of_Solutions_of (Segm (A,N,(Seg (width A))))
set L = (len A) |-> (0. K);
set C = ColVec2Mx ((len A) |-> (0. K));
A5: len ((len A) |-> (0. K)) = len A by CARD_1:def 7;
set S = Segm (A,N,(Seg (width A)));
A6: width (Segm (A,N,(Seg (width A)))) = card (Seg (width A)) by A2, MATRIX_0:23;
then A7: width A = width (Segm (A,N,(Seg (width A)))) by FINSEQ_1:57;
set SS = Space_of_Solutions_of (Segm (A,N,(Seg (width A))));
len (Segm (A,N,(Seg (width A)))) = card N by MATRIX_0:def 2;
then A8: the carrier of (Space_of_Solutions_of (Segm (A,N,(Seg (width A))))) = Solutions_of ((Segm (A,N,(Seg (width A)))),((card N) |-> (0. K))) by A3, A6, Def5;
set SA = Space_of_Solutions_of A;
A9: the carrier of (Space_of_Solutions_of A) = Solutions_of (A,((len A) |-> (0. K))) by A3, Def5;
A10: ColVec2Mx ((len A) |-> (0. K)) = 0. (K,(len A),1) by Th32;
len (ColVec2Mx ((len A) |-> (0. K))) = len ((len A) |-> (0. K)) by MATRIX_0:def 2;
then A11: dom (ColVec2Mx ((len A) |-> (0. K))) = dom A by A5, FINSEQ_3:29;
A12: dom A = Seg (len A) by FINSEQ_1:def 3;
then A13: Seg (len A) <> {} by A1, A2;
then A14: width (ColVec2Mx ((len A) |-> (0. K))) = 1 by Th26;
then A15: card (Seg (width (ColVec2Mx ((len A) |-> (0. K))))) = 1 by FINSEQ_1:57;

now :: thesis: for k, l being Nat st [k,l] in Indices (Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))))) holds
(Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))))) * (k,l) = (0. (K,(card N),1)) * (k,l)

A16: rng (Sgm (Seg 1)) = Seg 1 by FINSEQ_1:def 14;
A17: Indices (Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))))) = Indices (0. (K,(card N),1)) by A15, MATRIX_0:26;
let k, l be Nat; :: thesis: ( [k,l] in Indices (Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))))) implies (Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))))) * (k,l) = (0. (K,(card N),1)) * (k,l) )
assume A18: [k,l] in Indices (Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))))) ; :: thesis: (Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))))) * (k,l) = (0. (K,(card N),1)) * (k,l)
reconsider kk = k, ll = l as Element of NAT by ORDINAL1:def 12;
( [:N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))):] c= Indices (ColVec2Mx ((len A) |-> (0. K))) & rng (Sgm N) = N ) by A1, A11, FINSEQ_1:def 14, ZFMISC_1:95;
then A19: [((Sgm N) . kk),((Sgm (Seg (width (ColVec2Mx ((len A) |-> (0. K)))))) . ll)] in Indices (ColVec2Mx ((len A) |-> (0. K))) by A14, A18, A16, MATRIX13:17;
thus (Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))))) * (k,l) = (ColVec2Mx ((len A) |-> (0. K))) * (((Sgm N) . kk),((Sgm (Seg (width (ColVec2Mx ((len A) |-> (0. K)))))) . ll)) by A18, MATRIX13:def 1
.= 0. K by A10, A19, MATRIX_3:1
.= (0. (K,(card N),1)) * (k,l) by A18, A17, MATRIX_3:1 ; :: thesis: verum

end;

then A20: Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K)))))) = 0. (K,(card N),1) by A15, MATRIX_0:27
.= ColVec2Mx ((card N) |-> (0. K)) by Th32 ;

now :: thesis: for i being Nat st i in (dom A) \ N holds
( Line (A,i) = (width A) |-> (0. K) & Line ((ColVec2Mx ((len A) |-> (0. K))),i) = (width (ColVec2Mx ((len A) |-> (0. K)))) |-> (0. K) )

let i be Nat; :: thesis: ( i in (dom A) \ N implies ( Line (A,i) = (width A) |-> (0. K) & Line ((ColVec2Mx ((len A) |-> (0. K))),i) = (width (ColVec2Mx ((len A) |-> (0. K)))) |-> (0. K) ) )
assume A21: i in (dom A) \ N ; :: thesis: ( Line (A,i) = (width A) |-> (0. K) & Line ((ColVec2Mx ((len A) |-> (0. K))),i) = (width (ColVec2Mx ((len A) |-> (0. K)))) |-> (0. K) )
A22: i in dom A by A21, XBOOLE_0:def 5;
then Line ((ColVec2Mx ((len A) |-> (0. K))),i) = (ColVec2Mx ((len A) |-> (0. K))) . i by A5, A12, MATRIX_0:52
.= ((len A) |-> ((width (ColVec2Mx ((len A) |-> (0. K)))) |-> (0. K))) . i by A10, A13, Th26
.= (width (ColVec2Mx ((len A) |-> (0. K)))) |-> (0. K) by A12, A22, FINSEQ_2:57 ;
hence ( Line (A,i) = (width A) |-> (0. K) & Line ((ColVec2Mx ((len A) |-> (0. K))),i) = (width (ColVec2Mx ((len A) |-> (0. K)))) |-> (0. K) ) by A4, A21; :: thesis: verum

end;

then Solutions_of (A,(ColVec2Mx ((len A) |-> (0. K)))) = Solutions_of ((Segm (A,N,(Seg (width A)))),(Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K)))))))) by A1, A2, A11, Th45;
hence Space_of_Solutions_of A = Space_of_Solutions_of (Segm (A,N,(Seg (width A)))) by A20, A7, A9, A8, VECTSP_4:29; :: thesis: verum