let Omega be non empty set ; :: thesis: for Sigma being SigmaField of Omega
for P being Probability of Sigma
for C being non empty Subset-Family of Omega st ( for A being set holds
( A in C iff ex A1, A2 being set st
( A1 in Sigma & A2 in Sigma & A1 c= A & A c= A2 & P . (A2 \ A1) = 0 ) ) ) holds
C = COM (Sigma,P)
let Sigma be SigmaField of Omega; :: thesis: for P being Probability of Sigma
for C being non empty Subset-Family of Omega st ( for A being set holds
( A in C iff ex A1, A2 being set st
( A1 in Sigma & A2 in Sigma & A1 c= A & A c= A2 & P . (A2 \ A1) = 0 ) ) ) holds
C = COM (Sigma,P)
let P be Probability of Sigma; :: thesis: for C being non empty Subset-Family of Omega st ( for A being set holds
( A in C iff ex A1, A2 being set st
( A1 in Sigma & A2 in Sigma & A1 c= A & A c= A2 & P . (A2 \ A1) = 0 ) ) ) holds
C = COM (Sigma,P)
let C be non empty Subset-Family of Omega; :: thesis: ( ( for A being set holds
( A in C iff ex A1, A2 being set st
( A1 in Sigma & A2 in Sigma & A1 c= A & A c= A2 & P . (A2 \ A1) = 0 ) ) ) implies C = COM (Sigma,P) )
assume A1: for A being set holds
( A in C iff ex A1, A2 being set st
( A1 in Sigma & A2 in Sigma & A1 c= A & A c= A2 & P . (A2 \ A1) = 0 ) ) ; :: thesis: C = COM (Sigma,P)

now

let A be set ; :: thesis: ( A in C iff A in COM (Sigma,P) )
( A in C iff ex A1, A2 being set st
( A1 in Sigma & A2 in Sigma & A1 c= A & A c= A2 & P . (A2 \ A1) = 0 ) ) by A1;
hence ( A in C iff A in COM (Sigma,P) ) by Th36; :: thesis: verum

end;

hence C = COM (Sigma,P) by TARSKI:1; :: thesis: verum