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 5; :: 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_2:31, PROB_2:35; :: thesis:

end;

suppose u <> {} ; :: thesis:

then u = [#] Sigma by A11, TARSKI:def 2;
hence u,v are_independent_respect_to P by PROB_2:31, PROB_2:36; :: 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_2:31; :: 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

consider X being set ;
defpred S₁[ set ] means ex i being set st
( i in I & $1 in F . i );
let x be set ; :: according to TARSKI:def 3 :: thesis:
A16: dom F c= I \ ({} I) ;
assume x in Union (F | (I \ ({} I))) ; :: thesis:
then x in union (rng F) by A16, RELAT_1:97;
then consider y being set such that
A17: x in y and
A18: y in rng F by TARSKI:def 4;
consider i being set such that
A19: i in dom F and
A20: y = F . i by A18, FUNCT_1:def 5;
A21: x in finSigmaFields F,I

proof

reconsider x = x as Subset of Omega by A17, A20;
reconsider Fi = F . i as SigmaField of Omega by A19, Def2;
A22: ( sigma Fi c= Fi & Fi c= sigma Fi ) by PROB_1:def 14;
i in I by A19;
then for z being set 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;
A23: dom (F | {i}) = (dom F) /\ {i} by RELAT_1:90
.= {i} by A19, ZFMISC_1:52 ;
then rng (F | {i}) = {((F | {i}) . i)} by FUNCT_1:14;
then A24: union (rng (F | {i})) = (F | {i}) . i by ZFMISC_1:31;
i in dom (F | {i}) by A23, TARSKI:def 1;
then Union (F | {i}) = F . i by A24, FUNCT_1:70;
then sigUn F,E = F . i by A22, XBOOLE_0:def 10;
hence x in finSigmaFields F,I by A17, A20, Def10; :: thesis:

end;

finSigmaFields F,I c= sigma (finSigmaFields F,I) by PROB_1:def 14;
hence x in sigma (finSigmaFields F,I) by A21; :: thesis:

end;

then A25: ( T c= sigma T & sigUn F,(I \ ({} I)) c= sigma (finSigmaFields F,I) ) by PROB_1:def 14;
A26: 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
A27: x in finSigmaFields F,I and
A28: y in finSigmaFields F,I ; :: thesis:
consider E1 being finite Subset of I such that
A29: x in sigUn F,E1 by A27, Def10;
consider E2 being finite Subset of I such that
A30: y in sigUn F,E2 by A28, Def10;
A31: 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 ;
A32: Union (F | E1) c= Union (F | E3)

proof

end;

end;

then A36: y in sigUn F,(E2 \/ E1) by A30;
x in sigUn F,(E1 \/ E2) by A29, A31;
then x /\ y in sigUn F,(E1 \/ E2) by A36, FINSUB_1:def 2;
hence x /\ y in finSigmaFields F,I by Def10; :: thesis:

end;

then A37: finSigmaFields F,I is intersection_stable by FINSUB_1:def 2;
let u, v be Event of Sigma; :: thesis:
A38: not finSigmaFields F,I is empty

proof

consider E being finite Subset of I;
{} in sigUn F,E by PROB_1:10;
hence not finSigmaFields F,I is empty by Def10; :: thesis:

end;

A39: tailSigmaField F,I c= Sigma

proof

set Ie = I \ ({} I);
let x be set ; :: according to TARSKI:def 3 :: thesis:
assume A40: x in tailSigmaField F,I ; :: thesis:
Union (F | (I \ ({} I))) c= Sigma

proof

let y be set ; :: 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 A41: sigma (Union (F | (I \ ({} I)))) c= Sigma by PROB_1:def 14;
sigUn F,(I \ ({} I)) in futSigmaFields F,I by Def7;
then x in sigma (Union (F | (I \ ({} I)))) by A40, SETFAM_1:def 1;
hence x in Sigma by A41; :: thesis:

end;

A42: finSigmaFields F,I c= Sigma

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in finSigmaFields F,I ; :: thesis:
then consider E being finite Subset of I such that
A43: x in sigUn F,E by Def10;
Union (F | E) c= Sigma

proof

let y be set ; :: 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 14;
hence x in Sigma by A43; :: 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, A25, A38, A42, A37, A39, Th10; :: thesis:

end;

assume a in tailSigmaField F,I ; :: thesis:
then a,a are_independent_respect_to P by A1, A26;
then P . (a /\ a) = (P . a) * (P . a) by PROB_2:def 5;
hence ( P . a = 0 or P . a = 1 ) by Th2; :: thesis: