let b, m be FinSequence of INT ; :: thesis: ( len b = len m & ( for i being Nat st i in Seg (len b) holds

b . i <> 0 ) & m . 1 = 1 implies for k being Element of NAT st 1 <= k & k <= (len b) - 1 & ( for i being Nat st 1 <= i & i <= k holds

m . (i + 1) = (m . i) * (b . i) ) holds

m . (k + 1) <> 0 )

assume len b = len m ; :: thesis: ( ex i being Nat st

( i in Seg (len b) & not b . i <> 0 ) or not m . 1 = 1 or for k being Element of NAT st 1 <= k & k <= (len b) - 1 & ( for i being Nat st 1 <= i & i <= k holds

m . (i + 1) = (m . i) * (b . i) ) holds

m . (k + 1) <> 0 )

assume A1: ( ( for i being Nat st i in Seg (len b) holds

b . i <> 0 ) & m . 1 = 1 ) ; :: thesis: for k being Element of NAT st 1 <= k & k <= (len b) - 1 & ( for i being Nat st 1 <= i & i <= k holds

m . (i + 1) = (m . i) * (b . i) ) holds

m . (k + 1) <> 0

defpred S_{1}[ Nat] means ( 1 <= $1 & $1 <= (len b) - 1 & ( for i being Nat st 1 <= i & i <= $1 holds

m . (i + 1) = (m . i) * (b . i) ) implies m . ($1 + 1) <> 0 );

reconsider I0 = 0 as Element of NAT ;

A2: S_{1}[ 0 ]
;

A3: for k being Nat st S_{1}[k] holds

S_{1}[k + 1]
_{1}[k]
from NAT_1:sch 2(A2, A3);

hence for k being Element of NAT st 1 <= k & k <= (len b) - 1 & ( for i being Nat st 1 <= i & i <= k holds

m . (i + 1) = (m . i) * (b . i) ) holds

m . (k + 1) <> 0 ; :: thesis: verum

b . i <> 0 ) & m . 1 = 1 implies for k being Element of NAT st 1 <= k & k <= (len b) - 1 & ( for i being Nat st 1 <= i & i <= k holds

m . (i + 1) = (m . i) * (b . i) ) holds

m . (k + 1) <> 0 )

assume len b = len m ; :: thesis: ( ex i being Nat st

( i in Seg (len b) & not b . i <> 0 ) or not m . 1 = 1 or for k being Element of NAT st 1 <= k & k <= (len b) - 1 & ( for i being Nat st 1 <= i & i <= k holds

m . (i + 1) = (m . i) * (b . i) ) holds

m . (k + 1) <> 0 )

assume A1: ( ( for i being Nat st i in Seg (len b) holds

b . i <> 0 ) & m . 1 = 1 ) ; :: thesis: for k being Element of NAT st 1 <= k & k <= (len b) - 1 & ( for i being Nat st 1 <= i & i <= k holds

m . (i + 1) = (m . i) * (b . i) ) holds

m . (k + 1) <> 0

defpred S

m . (i + 1) = (m . i) * (b . i) ) implies m . ($1 + 1) <> 0 );

reconsider I0 = 0 as Element of NAT ;

A2: S

A3: for k being Nat st S

S

proof

for k being Nat holds S
let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

assume A4: S_{1}[k]
; :: thesis: S_{1}[k + 1]

assume A5: ( 1 <= k + 1 & k + 1 <= (len b) - 1 & ( for i being Nat st 1 <= i & i <= k + 1 holds

m . (i + 1) = (m . i) * (b . i) ) ) ; :: thesis: m . ((k + 1) + 1) <> 0

A6: k <= k + 1 by NAT_1:12;

end;assume A4: S

assume A5: ( 1 <= k + 1 & k + 1 <= (len b) - 1 & ( for i being Nat st 1 <= i & i <= k + 1 holds

m . (i + 1) = (m . i) * (b . i) ) ) ; :: thesis: m . ((k + 1) + 1) <> 0

A6: k <= k + 1 by NAT_1:12;

per cases
( k = 0 or k <> 0 )
;

end;

suppose A7:
k = 0
; :: thesis: m . ((k + 1) + 1) <> 0

A8: m . ((k + 1) + 1) =
(m . 1) * (b . 1)
by A5, A7

.= b . 1 by A1 ;

((len b) - 1) + 0 <= ((len b) - 1) + 1 by XREAL_1:7;

then k + 1 <= len b by A5, XXREAL_0:2;

then ( 1 <= 1 & 1 <= len b ) by A5, XXREAL_0:2;

then 1 in Seg (len b) ;

hence m . ((k + 1) + 1) <> 0 by A8, A1; :: thesis: verum

end;.= b . 1 by A1 ;

((len b) - 1) + 0 <= ((len b) - 1) + 1 by XREAL_1:7;

then k + 1 <= len b by A5, XXREAL_0:2;

then ( 1 <= 1 & 1 <= len b ) by A5, XXREAL_0:2;

then 1 in Seg (len b) ;

hence m . ((k + 1) + 1) <> 0 by A8, A1; :: thesis: verum

suppose A9:
k <> 0
; :: thesis: m . ((k + 1) + 1) <> 0

end;

A10: now :: thesis: for i being Nat st 1 <= i & i <= k holds

m . (i + 1) = (m . i) * (b . i)

thus
m . ((k + 1) + 1) <> 0
:: thesis: verumm . (i + 1) = (m . i) * (b . i)

let i be Nat; :: thesis: ( 1 <= i & i <= k implies m . (i + 1) = (m . i) * (b . i) )

assume ( 1 <= i & i <= k ) ; :: thesis: m . (i + 1) = (m . i) * (b . i)

then ( 1 <= i & i <= k + 1 ) by NAT_1:12;

hence m . (i + 1) = (m . i) * (b . i) by A5; :: thesis: verum

end;assume ( 1 <= i & i <= k ) ; :: thesis: m . (i + 1) = (m . i) * (b . i)

then ( 1 <= i & i <= k + 1 ) by NAT_1:12;

hence m . (i + 1) = (m . i) * (b . i) by A5; :: thesis: verum

proof

A11:
(k + 1) + 1 <= ((len b) - 1) + 1
by A5, XREAL_1:6;

A12: k + 1 <= (k + 1) + 1 by NAT_1:12;

A13: 1 <= k + 1 by NAT_1:12;

k + 1 <= len b by A12, A11, XXREAL_0:2;

then k + 1 in Seg (len b) by A13;

then A14: b . (k + 1) <> 0 by A1;

m . ((k + 1) + 1) = (m . (k + 1)) * (b . (k + 1)) by A5;

hence m . ((k + 1) + 1) <> 0 by A14, A10, A4, A5, A6, A9, NAT_1:14, XCMPLX_1:6, XXREAL_0:2; :: thesis: verum

end;A12: k + 1 <= (k + 1) + 1 by NAT_1:12;

A13: 1 <= k + 1 by NAT_1:12;

k + 1 <= len b by A12, A11, XXREAL_0:2;

then k + 1 in Seg (len b) by A13;

then A14: b . (k + 1) <> 0 by A1;

m . ((k + 1) + 1) = (m . (k + 1)) * (b . (k + 1)) by A5;

hence m . ((k + 1) + 1) <> 0 by A14, A10, A4, A5, A6, A9, NAT_1:14, XCMPLX_1:6, XXREAL_0:2; :: thesis: verum

hence for k being Element of NAT st 1 <= k & k <= (len b) - 1 & ( for i being Nat st 1 <= i & i <= k holds

m . (i + 1) = (m . i) * (b . i) ) holds

m . (k + 1) <> 0 ; :: thesis: verum