let Omega, I be non empty set ; :: thesis:
let Sigma be SigmaField of Omega; :: thesis:
let P be Probability of Sigma; :: thesis:
let a be Element of Sigma; :: thesis:
let F be ManySortedSigmaField of I,Sigma; :: thesis:
reconsider T = tailSigmaField (F,I) as SigmaField of Omega by Th15;
set Ie = I \ ({} I);
A1: ( a in tailSigmaField (F,I) implies a in sigUn (F,(I \ ({} I))) )

proof

assume A2: a in tailSigmaField (F,I) ; :: thesis:
sigUn (F,(I \ ({} I))) in futSigmaFields (F,I) by Def7;
hence a in sigUn (F,(I \ ({} I))) by A2, SETFAM_1:def 1; :: thesis:

end;

assume A3: F is_independent_wrt P ; :: thesis:
A4: for E being finite Subset of I
for p, q being Event of Sigma st p in sigUn (F,E) & q in tailSigmaField (F,I) holds
p,q are_independent_respect_to P

proof

let E be finite Subset of I; :: thesis:
let p, q be Event of Sigma; :: thesis:
assume that
A5: p in sigUn (F,E) and
A6: q in tailSigmaField (F,I) ; :: thesis:
for E being finite Subset of I holds q in sigUn (F,(I \ E))

proof

let E be finite Subset of I; :: thesis:
sigUn (F,(I \ E)) in futSigmaFields (F,I) by Def7;
hence q in sigUn (F,(I \ E)) by A6, SETFAM_1:def 1; :: thesis:

end;

then A7: q in sigUn (F,(I \ E)) ;

per cases ;

suppose A8: ( E <> {} & I \ E <> {} ) ; :: thesis:

then reconsider E = E as non empty Subset of I ;
reconsider IE = I \ E as non empty Subset of I by A8;
E misses IE by XBOOLE_1:79;
then P . (p /\ q) = (P . p) * (P . q) by A3, A5, A7, Th14;
hence p,q are_independent_respect_to P by PROB_2:def 4; :: thesis:

end;

suppose A9: ( not E <> {} or not I \ E <> {} ) ; :: thesis:

reconsider e = {} as Subset of I by XBOOLE_1:2;
A10: for u, v being Event of Sigma st u in {{},Omega} holds
u,v are_independent_respect_to P

proof

let u, v be Event of Sigma; :: thesis:
assume A11: u in {{},Omega} ; :: thesis:

per cases ;

suppose u = {} ; :: thesis:

hence u,v are_independent_respect_to P by PROB_2:19, PROB_2:23; :: thesis:

end;

suppose u <> {} ; :: thesis:

then u = [#] Sigma by A11, TARSKI:def 2;
hence u,v are_independent_respect_to P by PROB_2:19, PROB_2:24; :: thesis:

end;

Union (F | {}) = {} by ZFMISC_1:2;
then A12: sigUn (F,e) = {{},Omega} by Th3;
p,q are_independent_respect_to P

proof

per cases ;

suppose E = {} ; :: thesis:

hence p,q are_independent_respect_to P by A5, A12, A10; :: thesis:

end;

suppose E <> {} ; :: thesis:

hence p,q are_independent_respect_to P by A7, A9, A12, A10, PROB_2:19; :: thesis:

end;

hence p,q are_independent_respect_to P ; :: thesis:

end;

A13: for p, q being Event of Sigma st p in finSigmaFields (F,I) & q in tailSigmaField (F,I) holds
p,q are_independent_respect_to P

proof

let p, q be Event of Sigma; :: thesis:
assume that
A14: p in finSigmaFields (F,I) and
A15: q in tailSigmaField (F,I) ; :: thesis:
ex E being finite Subset of I st p in sigUn (F,E) by A14, Def10;
hence p,q are_independent_respect_to P by A4, A15; :: thesis:

end;

Union (F | (I \ ({} I))) c= sigma (finSigmaFields (F,I))

proof

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in Union (F | (I \ ({} I))) ; :: thesis:
then x in union (rng F) ;
then consider y being set such that
A16: x in y and
A17: y in rng F by TARSKI:def 4;
consider i being object such that
A18: i in dom F and
A19: y = F . i by A17, FUNCT_1:def 3;
A20: x in finSigmaFields (F,I)

proof

reconsider Fi = F . i as SigmaField of Omega by A18, Def2;
A21: ( sigma Fi c= Fi & Fi c= sigma Fi ) by PROB_1:def 9;
i in I by A18;
then for z being object st z in {i} holds
z in I by TARSKI:def 1;
then reconsider E = {i} as finite Subset of I by TARSKI:def 3;
A22: dom (F | {i}) = (dom F) /\ {i} by RELAT_1:61
.= {i} by A18, ZFMISC_1:46 ;
then rng (F | {i}) = {((F | {i}) . i)} by FUNCT_1:4;
then A23: union (rng (F | {i})) = (F | {i}) . i by ZFMISC_1:25;
i in dom (F | {i}) by A22, TARSKI:def 1;
then Union (F | {i}) = F . i by A23, FUNCT_1:47;
then sigUn (F,E) = F . i by A21;
hence x in finSigmaFields (F,I) by A16, A19, Def10; :: thesis:

end;

finSigmaFields (F,I) c= sigma (finSigmaFields (F,I)) by PROB_1:def 9;
hence x in sigma (finSigmaFields (F,I)) by A20; :: thesis:

end;

then A24: ( T c= sigma T & sigUn (F,(I \ ({} I))) c= sigma (finSigmaFields (F,I)) ) by PROB_1:def 9;
A25: for u, v being Event of Sigma st u in sigUn (F,(I \ ({} I))) & v in tailSigmaField (F,I) holds
u,v are_independent_respect_to P

proof

for x, y being set st x in finSigmaFields (F,I) & y in finSigmaFields (F,I) holds
x /\ y in finSigmaFields (F,I)

proof

let x, y be set ; :: thesis:
assume that
A26: x in finSigmaFields (F,I) and
A27: y in finSigmaFields (F,I) ; :: thesis:
consider E1 being finite Subset of I such that
A28: x in sigUn (F,E1) by A26, Def10;
consider E2 being finite Subset of I such that
A29: y in sigUn (F,E2) by A27, Def10;
A30: for z being set
for E1, E2 being finite Subset of I st z in sigUn (F,E1) holds
z in sigUn (F,(E1 \/ E2))

proof

let z be set ; :: thesis:
let E1, E2 be finite Subset of I; :: thesis:
reconsider E3 = E1 \/ E2 as finite Subset of I ;
A31: Union (F | E1) c= Union (F | E3)

proof

end;

end;

then A35: y in sigUn (F,(E2 \/ E1)) by A29;
x in sigUn (F,(E1 \/ E2)) by A28, A30;
then x /\ y in sigUn (F,(E1 \/ E2)) by A35, FINSUB_1:def 2;
hence x /\ y in finSigmaFields (F,I) by Def10; :: thesis:

end;

then A36: finSigmaFields (F,I) is intersection_stable by FINSUB_1:def 2;
let u, v be Event of Sigma; :: thesis:
A37: not finSigmaFields (F,I) is empty

proof

set E = the finite Subset of I;
{} in sigUn (F, the finite Subset of I) by PROB_1:4;
hence not finSigmaFields (F,I) is empty by Def10; :: thesis:

end;

A38: tailSigmaField (F,I) c= Sigma

proof

set Ie = I \ ({} I);
let x be object ; :: according to TARSKI:def 3 :: thesis:
assume A39: x in tailSigmaField (F,I) ; :: thesis:
Union (F | (I \ ({} I))) c= Sigma

proof

let y be object ; :: according to TARSKI:def 3 :: thesis:
assume y in Union (F | (I \ ({} I))) ; :: thesis:
then ex S being set st
( y in S & S in rng (F | (I \ ({} I))) ) by TARSKI:def 4;
hence y in Sigma ; :: thesis:

end;

then A40: sigma (Union (F | (I \ ({} I)))) c= Sigma by PROB_1:def 9;
sigUn (F,(I \ ({} I))) in futSigmaFields (F,I) by Def7;
then x in sigma (Union (F | (I \ ({} I)))) by A39, SETFAM_1:def 1;
hence x in Sigma by A40; :: thesis:

end;

A41: finSigmaFields (F,I) c= Sigma

proof

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in finSigmaFields (F,I) ; :: thesis:
then consider E being finite Subset of I such that
A42: x in sigUn (F,E) by Def10;
Union (F | E) c= Sigma

proof

let y be object ; :: according to TARSKI:def 3 :: thesis:
assume y in Union (F | E) ; :: thesis:
then ex S being set st
( y in S & S in rng (F | E) ) by TARSKI:def 4;
hence y in Sigma ; :: thesis:

end;

then sigma (Union (F | E)) c= Sigma by PROB_1:def 9;
hence x in Sigma by A42; :: thesis:

end;

assume ( u in sigUn (F,(I \ ({} I))) & v in tailSigmaField (F,I) ) ; :: thesis:
hence u,v are_independent_respect_to P by A13, A24, A37, A41, A36, A38, Th10; :: thesis:

end;

assume a in tailSigmaField (F,I) ; :: thesis:
then a,a are_independent_respect_to P by A1, A25;
then P . (a /\ a) = (P . a) * (P . a) by PROB_2:def 4;
hence ( P . a = 0 or P . a = 1 ) by Th2; :: thesis: