let D be non empty set ; :: thesis: for M being Matrix of D
for p being FinSequence of D * st len p = len M & ( for k being Nat st k >= 1 & k < len M holds
p . (k + 1) = (p . k) ^ (M . (k + 1)) ) holds
for j being Nat st j >= 1 & j < len p holds
for l being Nat st l in dom (p . j) holds
(p . j) . l = (p . (j + 1)) . l
let M be Matrix of D; :: thesis: for p being FinSequence of D * st len p = len M & ( for k being Nat st k >= 1 & k < len M holds
p . (k + 1) = (p . k) ^ (M . (k + 1)) ) holds
for j being Nat st j >= 1 & j < len p holds
for l being Nat st l in dom (p . j) holds
(p . j) . l = (p . (j + 1)) . l
let p be FinSequence of D * ; :: thesis: ( len p = len M & ( for k being Nat st k >= 1 & k < len M holds
p . (k + 1) = (p . k) ^ (M . (k + 1)) ) implies for j being Nat st j >= 1 & j < len p holds
for l being Nat st l in dom (p . j) holds
(p . j) . l = (p . (j + 1)) . l )
assume that
A1: len p = len M and
A2: for k being Nat st k >= 1 & k < len M holds
p . (k + 1) = (p . k) ^ (M . (k + 1)) ; :: thesis: for j being Nat st j >= 1 & j < len p holds
for l being Nat st l in dom (p . j) holds
(p . j) . l = (p . (j + 1)) . l
let j be Nat; :: thesis: ( j >= 1 & j < len p implies for l being Nat st l in dom (p . j) holds
(p . j) . l = (p . (j + 1)) . l )
assume that
A3: j >= 1 and
A4: j < len p ; :: thesis: for l being Nat st l in dom (p . j) holds
(p . j) . l = (p . (j + 1)) . l
A5: p . (j + 1) = (p . j) ^ (M . (j + 1)) by A1, A2, A3, A4;
let l be Nat; :: thesis: ( l in dom (p . j) implies (p . j) . l = (p . (j + 1)) . l )
assume l in dom (p . j) ; :: thesis: (p . j) . l = (p . (j + 1)) . l
hence (p . j) . l = (p . (j + 1)) . l by A5, FINSEQ_1:def 7; :: thesis: verum