let X be set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for T being N_Sub_set_fam of X st ( for A being set st A in T holds
A in S ) holds
( union T in S & meet T in S )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for T being N_Sub_set_fam of X st ( for A being set st A in T holds
A in S ) holds
( union T in S & meet T in S )
let M be sigma_Measure of S; :: thesis: for T being N_Sub_set_fam of X st ( for A being set st A in T holds
A in S ) holds
( union T in S & meet T in S )
let T be N_Sub_set_fam of X; :: thesis: ( ( for A being set st A in T holds
A in S ) implies ( union T in S & meet T in S ) )
assume A1: for A being set st A in T holds
A in S ; :: thesis: ( union T in S & meet T in S )
T c= S

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in T or x in S )
assume x in T ; :: thesis: x in S
hence x in S by A1; :: thesis: verum

end;

hence ( union T in S & meet T in S ) by Def9, Th49; :: thesis: verum