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 in dom p & j in dom p & i <= j holds
dom (p . i) c= dom (p . 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 in dom p & j in dom p & i <= j holds
dom (p . i) c= dom (p . 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 in dom p & j in dom p & i <= j holds
dom (p . i) c= dom (p . 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 in dom p & j in dom p & i <= j holds
dom (p . i) c= dom (p . j)
let i, j be Nat; :: thesis: ( i in dom p & j in dom p & i <= j implies dom (p . i) c= dom (p . j) )
assume that
A4: i in dom p and
A5: j in dom p and
A6: i <= j ; :: thesis: dom (p . i) c= dom (p . j)
A7: len (p . j) = j * (width M) by A1, A2, A3, A5, Th29;
len (p . i) = i * (width M) by A1, A2, A3, A4, Th29;
then len (p . i) <= len (p . j) by A6, A7, NAT_1:4;
then Seg (len (p . i)) c= Seg (len (p . j)) by FINSEQ_1:5;
then dom (p . i) c= Seg (len (p . j)) by FINSEQ_1:def 3;
hence dom (p . i) c= dom (p . j) by FINSEQ_1:def 3; :: thesis: verum