let i be Nat; :: thesis: for K being Field
for a being Element of K
for A being Matrix of K
for G being FinSequence_of_Matrix of K st i in dom A & width A = width (DelLine (A,i)) holds
DelLine ((block_diagonal ((<*A*> ^ G),a)),i) = block_diagonal ((<*(DelLine (A,i))*> ^ G),a)
let K be Field; :: thesis: for a being Element of K
for A being Matrix of K
for G being FinSequence_of_Matrix of K st i in dom A & width A = width (DelLine (A,i)) holds
DelLine ((block_diagonal ((<*A*> ^ G),a)),i) = block_diagonal ((<*(DelLine (A,i))*> ^ G),a)
let a be Element of K; :: thesis: for A being Matrix of K
for G being FinSequence_of_Matrix of K st i in dom A & width A = width (DelLine (A,i)) holds
DelLine ((block_diagonal ((<*A*> ^ G),a)),i) = block_diagonal ((<*(DelLine (A,i))*> ^ G),a)
let A be Matrix of K; :: thesis: for G being FinSequence_of_Matrix of K st i in dom A & width A = width (DelLine (A,i)) holds
DelLine ((block_diagonal ((<*A*> ^ G),a)),i) = block_diagonal ((<*(DelLine (A,i))*> ^ G),a)
let G be FinSequence_of_Matrix of K; :: thesis: ( i in dom A & width A = width (DelLine (A,i)) implies DelLine ((block_diagonal ((<*A*> ^ G),a)),i) = block_diagonal ((<*(DelLine (A,i))*> ^ G),a) )
assume that
A1: i in dom A and
A2: width A = width (DelLine (A,i)) ; :: thesis: DelLine ((block_diagonal ((<*A*> ^ G),a)),i) = block_diagonal ((<*(DelLine (A,i))*> ^ G),a)
A3: i in Seg (len A) by A1, FINSEQ_1:def 3;
set da = DelLine (A,i);
consider m being Nat such that
A4: len A = m + 1 and
A5: len (DelLine (A,i)) = m by A1, FINSEQ_3:113;
set bG = block_diagonal (G,a);
set DA = <*(DelLine (A,i))*>;
set AA = <*A*>;
set BG = <*(block_diagonal (G,a))*>;
set bAG = block_diagonal (<*A,(block_diagonal (G,a))*>,a);
set bdAG = block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a);
A6: Seg (len (block_diagonal (<*A,(block_diagonal (G,a))*>,a))) = dom (block_diagonal (<*A,(block_diagonal (G,a))*>,a)) by FINSEQ_1:def 3;
A7: len (block_diagonal (<*A,(block_diagonal (G,a))*>,a)) = Sum (Len (<*A*> ^ <*(block_diagonal (G,a))*>)) by Def5;
then A8: len (block_diagonal (<*A,(block_diagonal (G,a))*>,a)) = (len A) + (len (block_diagonal (G,a))) by Th16;
then len (block_diagonal (<*A,(block_diagonal (G,a))*>,a)) >= len A by NAT_1:11;
then A9: Seg (len A) c= Seg (len (block_diagonal (<*A,(block_diagonal (G,a))*>,a))) by FINSEQ_1:7;
A10: len (block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a)) = Sum (Len (<*(DelLine (A,i))*> ^ <*(block_diagonal (G,a))*>)) by Def5;
then A11: len (block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a)) = m + (len (block_diagonal (G,a))) by A5, Th16;
A12: len (block_diagonal (<*A,(block_diagonal (G,a))*>,a)) = (m + (len (block_diagonal (G,a)))) + 1 by A4, A8;
A13: len (block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a)) = (len (DelLine (A,i))) + (len (block_diagonal (G,a))) by A10, Th16;

A14: now

m + (len (block_diagonal (G,a))) <= len (block_diagonal (<*A,(block_diagonal (G,a))*>,a)) by A12, NAT_1:11;
then A15: Seg (m + (len (block_diagonal (G,a)))) c= Seg (len (block_diagonal (<*A,(block_diagonal (G,a))*>,a))) by FINSEQ_1:7;
reconsider da9 = DelLine (A,i) as Matrix of len (DelLine (A,i)), width (DelLine (A,i)),K by MATRIX_2:7;
reconsider A9 = A as Matrix of len A, width A,K by MATRIX_2:7;
let j be Nat; :: thesis: ( 1 <= j & j <= m + (len (block_diagonal (G,a))) implies (Del ((block_diagonal (<*A,(block_diagonal (G,a))*>,a)),i)) . j = (block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a)) . j )
assume that
A17: 1 <= j and
A18: j <= m + (len (block_diagonal (G,a))) ; :: thesis: (Del ((block_diagonal (<*A,(block_diagonal (G,a))*>,a)),i)) . j = (block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a)) . j
A19: j in Seg (m + (len (block_diagonal (G,a)))) by A17, A18, FINSEQ_1:3;
A20: 1 <= 1 + j by NAT_1:11;
j + 1 <= len (block_diagonal (<*A,(block_diagonal (G,a))*>,a)) by A12, A18, XREAL_1:9;
then A22: j + 1 in Seg (len (block_diagonal (<*A,(block_diagonal (G,a))*>,a))) by A20;

now

per cases ( j < i or j >= i ) ;

suppose A23: j < i ; :: thesis: (Del ((block_diagonal (<*A,(block_diagonal (G,a))*>,a)),i)) . j = (block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a)) . j

i <= len A by A3, FINSEQ_1:3;
then A24: j < len A by A23, XXREAL_0:2;
then A25: j <= m by A4, NAT_1:13;
then A26: j in dom (DelLine (A,i)) by A5, A17, FINSEQ_3:27;
A27: j in dom A by A17, A24, FINSEQ_3:27;
then A28: j in Seg m by A17, A25;
j in Seg (len A) by A17, A24, FINSEQ_1:3;
then A29: Line (A9,j) = A . j by MATRIX_2:10
.= da9 . j by A23, FINSEQ_3:119
.= Line ((DelLine (A,i)),j) by A5, A28, MATRIX_2:10 ;
thus (Del ((block_diagonal (<*A,(block_diagonal (G,a))*>,a)),i)) . j = (block_diagonal (<*A,(block_diagonal (G,a))*>,a)) . j by A23, FINSEQ_3:119
.= Line ((block_diagonal (<*A,(block_diagonal (G,a))*>,a)),j) by A7, A19, A15, MATRIX_2:10
.= (Line ((DelLine (A,i)),j)) ^ ((width (block_diagonal (G,a))) |-> a) by A27, A29, Th23
.= Line ((block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a)),j) by A26, Th23
.= (block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a)) . j by A5, A10, A13, A19, MATRIX_2:10 ; :: thesis: verum

end;

suppose A30: j >= i ; :: thesis: (Del ((block_diagonal (<*A,(block_diagonal (G,a))*>,a)),i)) . j = (block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a)) . j

then A31: (Del ((block_diagonal (<*A,(block_diagonal (G,a))*>,a)),i)) . j = (block_diagonal (<*A,(block_diagonal (G,a))*>,a)) . (j + 1) by A12, A9, A3, A6, A18, FINSEQ_3:120
.= Line ((block_diagonal (<*A,(block_diagonal (G,a))*>,a)),(j + 1)) by A7, A22, MATRIX_2:10 ;

now

per cases ( j + 1 <= len A or j + 1 > len A ) ;

suppose A32: j + 1 <= len A ; :: thesis: (Del ((block_diagonal (<*A,(block_diagonal (G,a))*>,a)),i)) . j = Line ((block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a)),j)

then A33: j + 1 in dom A by A20, FINSEQ_3:27;
A34: j <= m by A4, A32, XREAL_1:10;
then A35: j in Seg m by A17, FINSEQ_1:3;
A36: j in dom (DelLine (A,i)) by A5, A17, A34, FINSEQ_3:27;
j + 1 in Seg (len A) by A20, A32;
then Line (A9,(j + 1)) = A . (j + 1) by MATRIX_2:10
.= da9 . j by A1, A4, A30, A34, FINSEQ_3:120
.= Line ((DelLine (A,i)),j) by A5, A35, MATRIX_2:10 ;
hence (Del ((block_diagonal (<*A,(block_diagonal (G,a))*>,a)),i)) . j = (Line ((DelLine (A,i)),j)) ^ ((width (block_diagonal (G,a))) |-> a) by A31, A33, Th23
.= Line ((block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a)),j) by A36, Th23 ;
:: thesis: verum

end;

suppose A37: j + 1 > len A ; :: thesis: (Del ((block_diagonal (<*A,(block_diagonal (G,a))*>,a)),i)) . j = Line ((block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a)),j)

then reconsider jL = (j + 1) - (len A) as Element of NAT by NAT_1:21;
jL <> 0 by A37;
then A38: jL >= 1 by NAT_1:14;
jL + (len A) <= (len (block_diagonal (G,a))) + (len A) by A8, A12, A18, XREAL_1:9;
then jL <= len (block_diagonal (G,a)) by XREAL_1:10;
then A39: jL in dom (block_diagonal (G,a)) by A38, FINSEQ_3:27;
thus (Del ((block_diagonal (<*A,(block_diagonal (G,a))*>,a)),i)) . j = Line ((block_diagonal (<*A,(block_diagonal (G,a))*>,a)),(jL + (len A))) by A31
.= ((width (DelLine (A,i))) |-> a) ^ (Line ((block_diagonal (G,a)),jL)) by A2, A39, Th23
.= Line ((block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a)),(jL + (len (DelLine (A,i))))) by A39, Th23
.= Line ((block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a)),j) by A4, A5 ; :: thesis: verum

end;

end;

end;

hence (Del ((block_diagonal (<*A,(block_diagonal (G,a))*>,a)),i)) . j = (block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a)) . j by A5, A10, A13, A19, MATRIX_2:10; :: thesis: verum

end;

end;

end;

hence (Del ((block_diagonal (<*A,(block_diagonal (G,a))*>,a)),i)) . j = (block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a)) . j ; :: thesis: verum

end;

A40: block_diagonal ((<*(DelLine (A,i))*> ^ G),a) = block_diagonal (<*(DelLine (A,i)),(block_diagonal (G,a))*>,a) by Th36;
A41: block_diagonal ((<*A*> ^ G),a) = block_diagonal (<*A,(block_diagonal (G,a))*>,a) by Th36;
len (Del ((block_diagonal (<*A,(block_diagonal (G,a))*>,a)),i)) = m + (len (block_diagonal (G,a))) by A12, A9, A3, A6, FINSEQ_3:118;
hence DelLine ((block_diagonal ((<*A*> ^ G),a)),i) = block_diagonal ((<*(DelLine (A,i))*> ^ G),a) by A11, A41, A40, A14, FINSEQ_1:18; :: thesis: verum