let D be non-empty V26() ManySortedSet of NAT ; :: thesis: for P being Probability_sequence of Trivial-SigmaField_sequence D
for n being Nat holds (Product-Probability (P,D)) . n is Probability of Trivial-SigmaField ((Product_dom D) . n)
let P be Probability_sequence of Trivial-SigmaField_sequence D; :: thesis: for n being Nat holds (Product-Probability (P,D)) . n is Probability of Trivial-SigmaField ((Product_dom D) . n)
defpred S1[ Nat] means (Product-Probability (P,D)) . $1 is Probability of Trivial-SigmaField ((Product_dom D) . $1);
A1: (Product-Probability (P,D)) . 0 = P . 0 by Def13;
D . 0 = (Product_dom D) . 0 by Def10;
then A2: S1[ 0 ] by A1;
A3: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; :: thesis: ( S1[k] implies S1[k + 1] )
assume S1[k] ; :: thesis: S1[k + 1]
A4: (Product-Probability (P,D)) . (k + 1) = Product-Probability (((Product_dom D) . k),(D . (k + 1)),(modetrans (((Product-Probability (P,D)) . k),(Trivial-SigmaField ((Product_dom D) . k)))),(P . (k + 1))) by Def13;
(Product_dom D) . (k + 1) = [:((Product_dom D) . k),(D . (k + 1)):] by Def10;
hence S1[k + 1] by A4; :: thesis: verum
end;
for n being Nat holds S1[n] from NAT_1:sch 2(A2, A3);
 hence for n being Nat holds (Product-Probability (P,D)) . n is Probability of Trivial-SigmaField ((Product_dom D) . n) ; :: thesis: verum