let seq1, seq2 be ManySortedSet of NAT ; :: thesis: ( seq1 . 0 = P . 0 & ( for i being Nat holds seq1 . (i + 1) = Product-Probability (((Product_dom D) . i),(D . (i + 1)),(modetrans ((seq1 . i),(Trivial-SigmaField ((Product_dom D) . i)))),(P . (i + 1))) ) & seq2 . 0 = P . 0 & ( for i being Nat holds seq2 . (i + 1) = Product-Probability (((Product_dom D) . i),(D . (i + 1)),(modetrans ((seq2 . i),(Trivial-SigmaField ((Product_dom D) . i)))),(P . (i + 1))) ) implies seq1 = seq2 )
assume that
A3: seq1 . 0 = P . 0 and
A4: for i being Nat holds seq1 . (i + 1) = Product-Probability (((Product_dom D) . i),(D . (i + 1)),(modetrans ((seq1 . i),(Trivial-SigmaField ((Product_dom D) . i)))),(P . (i + 1))) and
A5: seq2 . 0 = P . 0 and
A6: for i being Nat holds seq2 . (i + 1) = Product-Probability (((Product_dom D) . i),(D . (i + 1)),(modetrans ((seq2 . i),(Trivial-SigmaField ((Product_dom D) . i)))),(P . (i + 1))) ; :: thesis: seq1 = seq2
A7: dom seq2 = NAT by PARTFUN1:def 2;
defpred S₁[ Nat] means seq1 . $1 = seq2 . $1;
A8: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A9: S₁[k] ; :: thesis: S₁[k + 1]
thus seq1 . (k + 1) = Product-Probability (((Product_dom D) . k),(D . (k + 1)),(modetrans ((seq2 . k),(Trivial-SigmaField ((Product_dom D) . k)))),(P . (k + 1))) by A9, A4
.= seq2 . (k + 1) by A6 ; :: thesis: verum

end;

A10: S₁[ 0 ] by A3, A5;
for n being Nat holds S₁[n] from NAT_1:sch 2(A10, A8);
then for n being object st n in dom seq1 holds
seq1 . n = seq2 . n ;
hence seq1 = seq2 by A7, FUNCT_1:2, PARTFUN1:def 2; :: thesis: verum