defpred S₁[ set , set ] means c₂ is SigmaField of (S . S);
set X = union (rng S);
A1: for n being Element of NAT ex y being Element of bool (bool (union (rng S))) st S₁[n,y]

proof

let n be Element of NAT ; :: thesis: ex y being Element of bool (bool (union (rng S))) st S₁[n,y]
set y = the SigmaField of (S . n);
dom S = NAT by PARTFUN1:def 2;
then S . n in rng S by FUNCT_1:3;
then bool (S . n) c= bool (union (rng S)) by ZFMISC_1:67, ZFMISC_1:74;
hence ex y being Element of bool (bool (union (rng S))) st S₁[n,y] ; :: thesis: verum

end;

consider F being Function of NAT,(bool (bool (union (rng S)))) such that
A2: for n being Element of NAT holds S₁[n,F . n] from FUNCT_2:sch 3(A1);
take F ; :: thesis: F is S -SigmaField_sequence-like
let n be Nat; :: according to RANDOM_3:def 8 :: thesis: F . n is SigmaField of (S . n)
n in NAT by ORDINAL1:def 12;
hence F . n is SigmaField of (S . n) by A2; :: thesis: verum