let X be set ; :: thesis: for S being SigmaField of X
for F being Function of NAT,S
for f being PartFunc of X,ExtREAL
for A being set
for r being Real st ( for n being Element of NAT holds F . n = A /\ (great_eq_dom (f,(R_EAL (r + (1 / (n + 1)))))) ) holds
A /\ (great_dom (f,(R_EAL r))) = union (rng F)
let S be SigmaField of X; :: thesis: for F being Function of NAT,S
for f being PartFunc of X,ExtREAL
for A being set
for r being Real st ( for n being Element of NAT holds F . n = A /\ (great_eq_dom (f,(R_EAL (r + (1 / (n + 1)))))) ) holds
A /\ (great_dom (f,(R_EAL r))) = union (rng F)
let F be Function of NAT,S; :: thesis: for f being PartFunc of X,ExtREAL
for A being set
for r being Real st ( for n being Element of NAT holds F . n = A /\ (great_eq_dom (f,(R_EAL (r + (1 / (n + 1)))))) ) holds
A /\ (great_dom (f,(R_EAL r))) = union (rng F)
let f be PartFunc of X,ExtREAL; :: thesis: for A being set
for r being Real st ( for n being Element of NAT holds F . n = A /\ (great_eq_dom (f,(R_EAL (r + (1 / (n + 1)))))) ) holds
A /\ (great_dom (f,(R_EAL r))) = union (rng F)
let A be set ; :: thesis: for r being Real st ( for n being Element of NAT holds F . n = A /\ (great_eq_dom (f,(R_EAL (r + (1 / (n + 1)))))) ) holds
A /\ (great_dom (f,(R_EAL r))) = union (rng F)
let r be Real; :: thesis: ( ( for n being Element of NAT holds F . n = A /\ (great_eq_dom (f,(R_EAL (r + (1 / (n + 1)))))) ) implies A /\ (great_dom (f,(R_EAL r))) = union (rng F) )
assume A1: for n being Element of NAT holds F . n = A /\ (great_eq_dom (f,(R_EAL (r + (1 / (n + 1)))))) ; :: thesis: A /\ (great_dom (f,(R_EAL r))) = union (rng F)
for x being set st x in A /\ (great_dom (f,(R_EAL r))) holds
x in union (rng F)

proof

let x be set ; :: thesis: ( x in A /\ (great_dom (f,(R_EAL r))) implies x in union (rng F) )
assume A3: x in A /\ (great_dom (f,(R_EAL r))) ; :: thesis: x in union (rng F)
then A4: x in A by XBOOLE_0:def 4;
A5: x in great_dom (f,(R_EAL r)) by A3, XBOOLE_0:def 4;
ex Y being set st
( x in Y & Y in rng F )

proof

reconsider x = x as Element of X by A3;
A7: x in dom f by A5, Def14;
A8: R_EAL r < f . x by A5, Def14;
ex m being Element of NAT st R_EAL (r + (1 / (m + 1))) <= f . x

proof

per cases ( f . x = +infty or not f . x = +infty ) ;

suppose A10: f . x = +infty ; :: thesis: ex m being Element of NAT st R_EAL (r + (1 / (m + 1))) <= f . x

take 1 ; :: thesis: R_EAL (r + (1 / (1 + 1))) <= f . x
thus R_EAL (r + (1 / (1 + 1))) <= f . x by A10, XXREAL_0:4; :: thesis: verum

end;

suppose not f . x = +infty ; :: thesis: ex m being Element of NAT st R_EAL (r + (1 / (m + 1))) <= f . x

then not +infty <= f . x by XXREAL_0:4;
then reconsider y1 = f . x as Real by A8, XXREAL_0:48;
consider m being Element of NAT such that
A13: 1 / (m + 1) < y1 - r by A8, Th13;
take m ; :: thesis: R_EAL (r + (1 / (m + 1))) <= f . x
thus R_EAL (r + (1 / (m + 1))) <= f . x by A13, XREAL_1:22; :: thesis: verum

end;

end;

end;

then consider m being Element of NAT such that
A14: R_EAL (r + (1 / (m + 1))) <= f . x ;
x in great_eq_dom (f,(R_EAL (r + (1 / (m + 1))))) by A7, A14, Def15;
then A16: x in A /\ (great_eq_dom (f,(R_EAL (r + (1 / (m + 1)))))) by A4, XBOOLE_0:def 4;
m in NAT ;
then A18: m in dom F by FUNCT_2:def 1;
take F . m ; :: thesis: ( x in F . m & F . m in rng F )
thus ( x in F . m & F . m in rng F ) by A1, A16, A18, FUNCT_1:def 5; :: thesis: verum

end;

hence x in union (rng F) by TARSKI:def 4; :: thesis: verum

end;

then A19: A /\ (great_dom (f,(R_EAL r))) c= union (rng F) by TARSKI:def 3;
for x being set st x in union (rng F) holds
x in A /\ (great_dom (f,(R_EAL r)))

proof

let x be set ; :: thesis: ( x in union (rng F) implies x in A /\ (great_dom (f,(R_EAL r))) )
assume x in union (rng F) ; :: thesis: x in A /\ (great_dom (f,(R_EAL r)))
then consider Y being set such that
A22: x in Y and
A23: Y in rng F by TARSKI:def 4;
consider m being Element of NAT such that
m in dom F and
A24: F . m = Y by A23, PARTFUN1:26;
A25: x in A /\ (great_eq_dom (f,(R_EAL (r + (1 / (m + 1)))))) by A1, A22, A24;
then A26: x in A by XBOOLE_0:def 4;
A27: x in great_eq_dom (f,(R_EAL (r + (1 / (m + 1))))) by A25, XBOOLE_0:def 4;
then A28: x in dom f by Def15;
A29: R_EAL (r + (1 / (m + 1))) <= f . x by A27, Def15;
reconsider x = x as Element of X by A22, A23;
R_EAL r < f . x

proof

now

r < r + (1 / (m + 1)) by XREAL_1:31, XREAL_1:141;
hence R_EAL r < f . x by A29, XXREAL_0:2; :: thesis: verum

end;

hence R_EAL r < f . x ; :: thesis: verum

end;

then x in great_dom (f,(R_EAL r)) by A28, Def14;
hence x in A /\ (great_dom (f,(R_EAL r))) by A26, XBOOLE_0:def 4; :: thesis: verum

end;

then union (rng F) c= A /\ (great_dom (f,(R_EAL r))) by TARSKI:def 3;
hence A /\ (great_dom (f,(R_EAL r))) = union (rng F) by A19, XBOOLE_0:def 10; :: thesis: verum