consider A being set such that
A1: A in S by PROB_1:4;
reconsider A = A as Subset of X by A1;
consider B, C being Subset of X such that
A2: ( B = A & C = A ) ;
A3: {A} c= S by A1, ZFMISC_1:31;
consider F being sequence of (bool X) such that
A4: rng F = {B,C} and
A5: ( F . 0 = B & ( for n being Nat st 0 < n holds
F . n = C ) ) by MEASURE1:19;
A6: rng F = {A} by A2, A4, ENUMSET1:29;
then A7: rng F c= S by A1, ZFMISC_1:31;
{A} is N_Sub_set_fam of X by A6, SUPINF_2:def 8;
then reconsider T = {A} as N_Measure_fam of S by A3, Def1;
reconsider F = F as sequence of S by A7, FUNCT_2:6;
take T ; :: thesis: T is non-decreasing
take F ; :: according to MEASURE2:def 2 :: thesis: ( T = rng F & ( for n being Nat holds F . n c= F . (n + 1) ) )
for n being Nat holds F . n c= F . (n + 1)

proof

let n be Nat; :: thesis: F . n c= F . (n + 1)
F . n = A by A2, A5, NAT_1:3;
hence F . n c= F . (n + 1) by A2, A5, NAT_1:3; :: thesis: verum

end;

hence ( T = rng F & ( for n being Nat holds F . n c= F . (n + 1) ) ) by A2, A4, ENUMSET1:29; :: thesis: verum