consider A being set such that
A1: A in S by PROB_1:10;
reconsider A = A as Subset of X by A1;
consider B, C being Subset of X such that
A2: ( B = A & C = A ) ;
consider F being Function of NAT ,(bool X) such that
A3: ( rng F = {B,C} & F . 0 = B & ( for n being Element of NAT st 0 < n holds
F . n = C ) ) by MEASURE1:40;
A4: ( rng F = {A} & F . 0 = A & ( for n being Element of NAT st 0 < n holds
F . n = A ) ) by A2, A3, ENUMSET1:69;
then rng F c= S by A1, ZFMISC_1:37;
then reconsider F = F as Function of NAT ,S by FUNCT_2:8;
A5: {A} is N_Sub_set_fam of X by A4, SUPINF_2:def 14;
{A} c= S by A1, ZFMISC_1:37;
then reconsider T = {A} as N_Measure_fam of S by A5, Def1;
take T ; :: thesis: T is non-decreasing
take F ; :: according to MEASURE2:def 2 :: thesis: ( T = rng F & ( for n being Element of NAT holds F . n c= F . (n + 1) ) )
for n being Element of NAT holds F . n c= F . (n + 1)

proof

let n be Element of NAT ; :: thesis: F . n c= F . (n + 1)
( F . n = A & F . (n + 1) = A ) by A2, A3, NAT_1:3;
hence F . n c= F . (n + 1) ; :: thesis: verum

end;

hence ( T = rng F & ( for n being Element of NAT holds F . n c= F . (n + 1) ) ) by A2, A3, ENUMSET1:69; :: thesis: verum