let x, y be Element of INT ; :: thesis: for b, m being non empty FinSequence of INT st 2 <= len b & ( for i, j being Nat st i in Seg (len b) & j in Seg (len b) & i <> j holds
b . i,b . j are_relative_prime ) & ( for i being Nat st i in Seg (len b) holds
x mod (b . i) = y mod (b . i) ) & m . 1 = 1 holds
for k being Element of NAT st 1 <= k & k <= len b & ( for i being Nat st 1 <= i & i <= k holds
m . (i + 1) = (m . i) * (b . i) ) holds
x mod (m . (k + 1)) = y mod (m . (k + 1))
let b, m be non empty FinSequence of INT ; :: thesis: ( 2 <= len b & ( for i, j being Nat st i in Seg (len b) & j in Seg (len b) & i <> j holds
b . i,b . j are_relative_prime ) & ( for i being Nat st i in Seg (len b) holds
x mod (b . i) = y mod (b . i) ) & m . 1 = 1 implies for k being Element of NAT st 1 <= k & k <= len b & ( for i being Nat st 1 <= i & i <= k holds
m . (i + 1) = (m . i) * (b . i) ) holds
x mod (m . (k + 1)) = y mod (m . (k + 1)) )
assume A1: 2 <= len b ; :: thesis: ( ex i, j being Nat st
( i in Seg (len b) & j in Seg (len b) & i <> j & not b . i,b . j are_relative_prime ) or ex i being Nat st
( i in Seg (len b) & not x mod (b . i) = y mod (b . i) ) or not m . 1 = 1 or for k being Element of NAT st 1 <= k & k <= len b & ( for i being Nat st 1 <= i & i <= k holds
m . (i + 1) = (m . i) * (b . i) ) holds
x mod (m . (k + 1)) = y mod (m . (k + 1)) )
assume A4: for i, j being Nat st i in Seg (len b) & j in Seg (len b) & i <> j holds
b . i,b . j are_relative_prime ; :: thesis: ( ex i being Nat st
( i in Seg (len b) & not x mod (b . i) = y mod (b . i) ) or not m . 1 = 1 or for k being Element of NAT st 1 <= k & k <= len b & ( for i being Nat st 1 <= i & i <= k holds
m . (i + 1) = (m . i) * (b . i) ) holds
x mod (m . (k + 1)) = y mod (m . (k + 1)) )
assume A2: for i being Nat st i in Seg (len b) holds
x mod (b . i) = y mod (b . i) ; :: thesis: ( not m . 1 = 1 or for k being Element of NAT st 1 <= k & k <= len b & ( for i being Nat st 1 <= i & i <= k holds
m . (i + 1) = (m . i) * (b . i) ) holds
x mod (m . (k + 1)) = y mod (m . (k + 1)) )
assume A3: m . 1 = 1 ; :: thesis: for k being Element of NAT st 1 <= k & k <= len b & ( for i being Nat st 1 <= i & i <= k holds
m . (i + 1) = (m . i) * (b . i) ) holds
x mod (m . (k + 1)) = y mod (m . (k + 1))
defpred S₁[ Nat] means ( 1 <= $1 & $1 <= len b & ( for i being Nat st 1 <= i & i <= $1 holds
m . (i + 1) = (m . i) * (b . i) ) implies x mod (m . ($1 + 1)) = y mod (m . ($1 + 1)) );
reconsider I0 = 0 as Element of NAT ;
P0: S₁[ 0 ] ;
P1: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume P11: S₁[k] ; :: thesis: S₁[k + 1]
assume P12: ( 1 <= k + 1 & k + 1 <= len b & ( for i being Nat st 1 <= i & i <= k + 1 holds
m . (i + 1) = (m . i) * (b . i) ) ) ; :: thesis: x mod (m . ((k + 1) + 1)) = y mod (m . ((k + 1) + 1))
P14: k <= k + 1 by NAT_1:12;

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

suppose P15: k = 0 ; :: thesis: x mod (m . ((k + 1) + 1)) = y mod (m . ((k + 1) + 1))

P16: m . ((k + 1) + 1) = (m . 1) * (b . 1) by P12, P15
.= b . 1 by A3 ;
( 1 <= 1 & 1 <= len b ) by NAT_1:14;
then P18: 1 in Seg (len b) ;
thus x mod (m . ((k + 1) + 1)) = y mod (m . ((k + 1) + 1)) by P16, P18, A2; :: thesis: verum

end;

suppose P181: k <> 0 ; :: thesis: x mod (m . ((k + 1) + 1)) = y mod (m . ((k + 1) + 1))

(k + 1) - 1 <= (len b) - 1 by P12, XREAL_1:9;
then P19: ( 1 <= k & k <= (len b) - 1 ) by P181, NAT_1:14;

XX1: now :: thesis: for i being Nat st 1 <= i & i <= k holds
m . (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 P12; :: thesis: verum

end;

P22: m . ((k + 1) + 1) = (m . (k + 1)) * (b . (k + 1)) by P12;
k + 1 in Seg (len b) by P12;
then P23: x mod (b . (k + 1)) = y mod (b . (k + 1)) by A2;
P24: m . (k + 1),b . (k + 1) are_relative_prime by LmTh5, XX1, P19, A1, A4, A3, P12;
( m . (k + 1) is Element of INT & b . (k + 1) is Element of INT ) by INT_1:def 2;
hence x mod (m . ((k + 1) + 1)) = y mod (m . ((k + 1) + 1)) by P22, P23, P11, NAT_1:14, P181, P12, P14, XXREAL_0:2, XX1, P24, LmTh7A; :: thesis: verum

end;

end;

end;

for k being Element of NAT holds S₁[k] from NAT_1:sch 1(P0, P1);
hence for k being Element of NAT st 1 <= k & k <= len b & ( for i being Nat st 1 <= i & i <= k holds
m . (i + 1) = (m . i) * (b . i) ) holds
x mod (m . (k + 1)) = y mod (m . (k + 1)) ; :: thesis: verum