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) = () |-> (0. K) ) holds
Space_of_Solutions_of A = Space_of_Solutions_of (Segm (A,N,(Seg ())))

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) = () |-> (0. K) ) holds
Space_of_Solutions_of A = Space_of_Solutions_of (Segm (A,N,(Seg ())))

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) = () |-> (0. K) ) implies Space_of_Solutions_of A = Space_of_Solutions_of (Segm (A,N,(Seg ()))) )

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) = () |-> (0. K) ; :: thesis: Space_of_Solutions_of A = Space_of_Solutions_of (Segm (A,N,(Seg ())))
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 ()));
A6: width (Segm (A,N,(Seg ()))) = card (Seg ()) by ;
then A7: width A = width (Segm (A,N,(Seg ()))) by FINSEQ_1:57;
set SS = Space_of_Solutions_of (Segm (A,N,(Seg ())));
len (Segm (A,N,(Seg ()))) = card N by MATRIX_0:def 2;
then A8: the carrier of (Space_of_Solutions_of (Segm (A,N,(Seg ())))) = Solutions_of ((Segm (A,N,(Seg ()))),((card N) |-> (0. K))) by A3, A6, Def5;
set SA = Space_of_Solutions_of A;
A9: the carrier of = Solutions_of (A,((len A) |-> (0. K))) by ;
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 ;
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 13;
A17: Indices (Segm ((ColVec2Mx ((len A) |-> (0. K))),N,(Seg (width (ColVec2Mx ((len A) |-> (0. K))))))) = Indices (0. (K,(card N),1)) by ;
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 ;
then A19: [((Sgm N) . kk),((Sgm (Seg (width (ColVec2Mx ((len A) |-> (0. K)))))) . ll)] in Indices (ColVec2Mx ((len A) |-> (0. K))) by ;
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
.= 0. K by
.= (0. (K,(card N),1)) * (k,l) by ; :: 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
.= ColVec2Mx ((card N) |-> (0. K)) by Th32 ;
now :: thesis: for i being Nat st i in (dom A) \ N holds
( Line (A,i) = () |-> (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) = () |-> (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) = () |-> (0. K) & Line ((ColVec2Mx ((len A) |-> (0. K))),i) = (width (ColVec2Mx ((len A) |-> (0. K)))) |-> (0. K) )
A22: i in dom A by ;
then Line ((ColVec2Mx ((len A) |-> (0. K))),i) = (ColVec2Mx ((len A) |-> (0. K))) . i by
.= ((len A) |-> ((width (ColVec2Mx ((len A) |-> (0. K)))) |-> (0. K))) . i by
.= (width (ColVec2Mx ((len A) |-> (0. K)))) |-> (0. K) by ;
hence ( Line (A,i) = () |-> (0. K) & Line ((ColVec2Mx ((len A) |-> (0. K))),i) = (width (ColVec2Mx ((len A) |-> (0. K)))) |-> (0. K) ) by ; :: thesis: verum
end;
then Solutions_of (A,(ColVec2Mx ((len A) |-> (0. K)))) = Solutions_of ((Segm (A,N,(Seg ()))),(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 ()))) by ; :: thesis: verum