let D be non empty set ; :: thesis: for M being Matrix of D
for p being FinSequence of D * st len p = len M & p . 1 = M . 1 & ( for k being Nat st k >= 1 & k < len M holds
p . (k + 1) = (p . k) ^ (M . (k + 1)) ) holds
for i, j being Nat st [i,j] in Indices M holds
( ((i - 1) * (width M)) + j in dom (p . (len M)) & M * (i,j) = (p . (len M)) . (((i - 1) * (width M)) + j) )
let M be Matrix of D; :: thesis: for p being FinSequence of D * st len p = len M & p . 1 = M . 1 & ( for k being Nat st k >= 1 & k < len M holds
p . (k + 1) = (p . k) ^ (M . (k + 1)) ) holds
for i, j being Nat st [i,j] in Indices M holds
( ((i - 1) * (width M)) + j in dom (p . (len M)) & M * (i,j) = (p . (len M)) . (((i - 1) * (width M)) + j) )
let p be FinSequence of D * ; :: thesis: ( len p = len M & p . 1 = M . 1 & ( for k being Nat st k >= 1 & k < len M holds
p . (k + 1) = (p . k) ^ (M . (k + 1)) ) implies for i, j being Nat st [i,j] in Indices M holds
( ((i - 1) * (width M)) + j in dom (p . (len M)) & M * (i,j) = (p . (len M)) . (((i - 1) * (width M)) + j) ) )
assume that
A1: len p = len M and
A2: p . 1 = M . 1 and
A3: for k being Nat st k >= 1 & k < len M holds
p . (k + 1) = (p . k) ^ (M . (k + 1)) ; :: thesis: for i, j being Nat st [i,j] in Indices M holds
( ((i - 1) * (width M)) + j in dom (p . (len M)) & M * (i,j) = (p . (len M)) . (((i - 1) * (width M)) + j) )
let i, j be Nat; :: thesis: ( [i,j] in Indices M implies ( ((i - 1) * (width M)) + j in dom (p . (len M)) & M * (i,j) = (p . (len M)) . (((i - 1) * (width M)) + j) ) )
assume A4: [i,j] in Indices M ; :: thesis: ( ((i - 1) * (width M)) + j in dom (p . (len M)) & M * (i,j) = (p . (len M)) . (((i - 1) * (width M)) + j) )
A5: ((i - 1) * (width M)) + j in dom (p . i) by A1, A2, A3, A4, Th35;
A6: M * (i,j) = (p . i) . (((i - 1) * (width M)) + j) by A1, A2, A3, A4, Th35;
A7: i in Seg (len M) by A4, MATRPROB:12;
then A8: len M <> 0 ;
A9: i <= len M by A7, FINSEQ_1:1;
len M >= 1 by A8, NAT_1:14;
then len M in Seg (len p) by A1, FINSEQ_1:1;
then A10: len M in dom p by FINSEQ_1:def 3;
A11: i in dom p by A1, A7, FINSEQ_1:def 3;
then dom (p . i) c= dom (p . (len M)) by A1, A2, A3, A9, A10, Th30;
hence ( ((i - 1) * (width M)) + j in dom (p . (len M)) & M * (i,j) = (p . (len M)) . (((i - 1) * (width M)) + j) ) by A1, A2, A3, A9, A11, A10, A5, A6, Th33; :: thesis: verum