consider A being set such that

A1: A in S by PROB_1:4;

reconsider A = A as Subset of X by A1;

A2: {A} c= S by A1, ZFMISC_1:31;

set B = A;

set C = A;

consider F being sequence of (bool X) such that

A3: rng F = {A,A} and

A4: ( F . 0 = A & ( for n being Nat st 0 < n holds

F . n = A ) ) by MEASURE1:19;

A5: rng F = {A} by A3, ENUMSET1:29;

then A6: rng F c= S by A1, ZFMISC_1:31;

{A} is N_Sub_set_fam of X by A5, SUPINF_2:def 8;

then reconsider T = {A} as N_Measure_fam of S by A2, Def1;

reconsider F = F as sequence of S by A6, FUNCT_2:6;

take T ; :: thesis: T is non-increasing

take F ; :: according to MEASURE2:def 3 :: thesis: ( T = rng F & ( for n being Element of NAT holds F . (n + 1) c= F . n ) )

for n being Element of NAT holds F . (n + 1) c= F . n

A1: A in S by PROB_1:4;

reconsider A = A as Subset of X by A1;

A2: {A} c= S by A1, ZFMISC_1:31;

set B = A;

set C = A;

consider F being sequence of (bool X) such that

A3: rng F = {A,A} and

A4: ( F . 0 = A & ( for n being Nat st 0 < n holds

F . n = A ) ) by MEASURE1:19;

A5: rng F = {A} by A3, ENUMSET1:29;

then A6: rng F c= S by A1, ZFMISC_1:31;

{A} is N_Sub_set_fam of X by A5, SUPINF_2:def 8;

then reconsider T = {A} as N_Measure_fam of S by A2, Def1;

reconsider F = F as sequence of S by A6, FUNCT_2:6;

take T ; :: thesis: T is non-increasing

take F ; :: according to MEASURE2:def 3 :: thesis: ( T = rng F & ( for n being Element of NAT holds F . (n + 1) c= F . n ) )

for n being Element of NAT holds F . (n + 1) c= F . n

proof

hence
( T = rng F & ( for n being Element of NAT holds F . (n + 1) c= F . n ) )
by A3, ENUMSET1:29; :: thesis: verum
let n be Element of NAT ; :: thesis: F . (n + 1) c= F . n

F . n = A by A4, NAT_1:3;

hence F . (n + 1) c= F . n by A4, NAT_1:3; :: thesis: verum

end;F . n = A by A4, NAT_1:3;

hence F . (n + 1) c= F . n by A4, NAT_1:3; :: thesis: verum