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_coprime ) & ( 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_coprime ) & ( 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_coprime ) 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 A2: for i, j being Nat st i in Seg (len b) & j in Seg (len b) & i <> j holds
b . i,b . j are_coprime ; :: 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 A3: 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 A4: 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 ;
A5: S₁[ 0 ] ;
A6: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A7: S₁[k] ; :: thesis: S₁[k + 1]
assume A8: ( 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))
A9: k <= k + 1 by NAT_1:12;

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

suppose A10: k = 0 ; :: thesis: x mod (m . ((k + 1) + 1)) = y mod (m . ((k + 1) + 1))

A11: m . ((k + 1) + 1) = (m . 1) * (b . 1) by A8, A10
.= b . 1 by A4 ;
( 1 <= 1 & 1 <= len b ) by NAT_1:14;
then A12: 1 in Seg (len b) ;
thus x mod (m . ((k + 1) + 1)) = y mod (m . ((k + 1) + 1)) by A11, A12, A3; :: thesis: verum

end;

suppose A13: k <> 0 ; :: thesis: x mod (m . ((k + 1) + 1)) = y mod (m . ((k + 1) + 1))

(k + 1) - 1 <= (len b) - 1 by A8, XREAL_1:9;
then A14: ( 1 <= k & k <= (len b) - 1 ) by A13, NAT_1:14;

A15: 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 A8; :: thesis: verum

end;

A16: m . ((k + 1) + 1) = (m . (k + 1)) * (b . (k + 1)) by A8;
k + 1 in Seg (len b) by A8;
then A17: x mod (b . (k + 1)) = y mod (b . (k + 1)) by A3;
A18: m . (k + 1),b . (k + 1) are_coprime by Lm16, A15, A14, A1, A2, A4, A8;
( 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 A16, A17, A7, A13, A8, A9, A15, A18, Lm18, NAT_1:14, XXREAL_0:2; :: thesis: verum

end;

end;

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A5, A6);
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