let a, b be Integer; :: thesis: for A, B being sequence of NAT st A . 0 = |.a.| & B . 0 = |.b.| & ( for i being Nat holds
( A . (i + 1) = B . i & B . (i + 1) = (A . i) mod (B . i) ) ) holds
{ i where i is Nat : B . i = 0 } is non empty Subset of NAT

let A, B be sequence of NAT; :: thesis: ( A . 0 = |.a.| & B . 0 = |.b.| & ( for i being Nat holds
( A . (i + 1) = B . i & B . (i + 1) = (A . i) mod (B . i) ) ) implies { i where i is Nat : B . i = 0 } is non empty Subset of NAT )

assume A1: ( A . 0 = |.a.| & B . 0 = |.b.| & ( for i being Nat holds
( A . (i + 1) = B . i & B . (i + 1) = (A . i) mod (B . i) ) ) ) ; :: thesis: { i where i is Nat : B . i = 0 } is non empty Subset of NAT
A2: for x being object st x in { i where i is Nat : B . i = 0 } holds
x in NAT
proof
let x be object ; :: thesis: ( x in { i where i is Nat : B . i = 0 } implies x in NAT )
assume x in { i where i is Nat : B . i = 0 } ; :: thesis:
then ex i being Nat st
( x = i & B . i = 0 ) ;
hence x in NAT by ORDINAL1:def 12; :: thesis: verum
end;
ex m0 being Nat st B . m0 = 0
proof
assume A3: for m0 being Nat holds not B . m0 = 0 ; :: thesis: contradiction
A4: for i being Nat holds B . (i + 1) <= (B . i) - 1
proof
let i be Nat; :: thesis: B . (i + 1) <= (B . i) - 1
A5: B . i <> 0 by A3;
B . (i + 1) = (A . i) mod (B . i) by A1;
then (B . (i + 1)) + 1 <= B . i by ;
then ((B . (i + 1)) + 1) - 1 <= (B . i) - 1 by XREAL_1:9;
hence B . (i + 1) <= (B . i) - 1 ; :: thesis: verum
end;
defpred S1[ Nat] means B . \$1 <= (B . 0) - \$1;
A6: S1[ 0 ] ;
A7: for i being Nat st S1[i] holds
S1[i + 1]
proof
let i be Nat; :: thesis: ( S1[i] implies S1[i + 1] )
assume A8: S1[i] ; :: thesis: S1[i + 1]
A9: B . (i + 1) <= (B . i) - 1 by A4;
(B . i) - 1 <= ((B . 0) - i) - 1 by ;
hence S1[i + 1] by ; :: thesis: verum
end;
A10: for i being Nat holds S1[i] from NAT_1:sch 2(A6, A7);
B . (B . 0) <= (B . 0) - (B . 0) by A10;
hence contradiction by A3, NAT_1:14; :: thesis: verum
end;
then consider m0 being Nat such that
A11: B . m0 = 0 ;
m0 in { i where i is Nat : B . i = 0 } by A11;
hence { i where i is Nat : B . i = 0 } is non empty Subset of NAT by ; :: thesis: verum