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 st ( for n being Element of NAT holds F . n = A /\ (great_dom f,(R_EAL (- n))) ) holds
A /\ (great_dom f,-infty ) = 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 st ( for n being Element of NAT holds F . n = A /\ (great_dom f,(R_EAL (- n))) ) holds
A /\ (great_dom f,-infty ) = union (rng F)
let F be Function of NAT ,S; :: thesis: for f being PartFunc of X, ExtREAL
for A being set st ( for n being Element of NAT holds F . n = A /\ (great_dom f,(R_EAL (- n))) ) holds
A /\ (great_dom f,-infty ) = union (rng F)
let f be PartFunc of X, ExtREAL ; :: thesis: for A being set st ( for n being Element of NAT holds F . n = A /\ (great_dom f,(R_EAL (- n))) ) holds
A /\ (great_dom f,-infty ) = union (rng F)
let A be set ; :: thesis: ( ( for n being Element of NAT holds F . n = A /\ (great_dom f,(R_EAL (- n))) ) implies A /\ (great_dom f,-infty ) = union (rng F) )
assume A1: for n being Element of NAT holds F . n = A /\ (great_dom f,(R_EAL (- n))) ; :: thesis: A /\ (great_dom f,-infty ) = union (rng F)
A2: A /\ (great_dom f,-infty ) c= union (rng F)

proof

for x being set st x in A /\ (great_dom f,-infty ) holds
x in union (rng F)

proof

let x be set ; :: thesis: ( x in A /\ (great_dom f,-infty ) implies x in union (rng F) )
assume A3: x in A /\ (great_dom f,-infty ) ; :: thesis: x in union (rng F)
then A4: ( x in A & x in great_dom f,-infty ) by XBOOLE_0:def 4;
then A5: x in dom f by Def14;
A6: -infty < f . x by A4, Def14;
ex n being Element of NAT st R_EAL (- n) < f . x

proof

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

suppose A7: f . x = +infty ; :: thesis: ex n being Element of NAT st R_EAL (- n) < f . x

take 0 ; :: thesis: R_EAL (- 0 ) < f . x
thus R_EAL (- 0 ) < f . x by A7; :: thesis: verum

end;

suppose not f . x = +infty ; :: thesis: ex n being Element of NAT st R_EAL (- n) < f . x

then not +infty <= f . x by XXREAL_0:4;
then reconsider y1 = f . x as Real by A6, XXREAL_0:48;
consider n1 being Element of NAT such that
A8: - n1 <= y1 by Th12;
n1 < n1 + 1 by NAT_1:13;
then A9: - (n1 + 1) < - n1 by XREAL_1:26;
reconsider m = n1 + 1 as Element of NAT ;
take m ; :: thesis: R_EAL (- m) < f . x
thus R_EAL (- m) < f . x by A8, A9, XXREAL_0:2; :: thesis: verum

end;

end;

end;

then consider n being Element of NAT such that
A10: R_EAL (- n) < f . x ;
reconsider x = x as Element of X by A3;
x in great_dom f,(R_EAL (- n)) by A5, A10, Def14;
then x in A /\ (great_dom f,(R_EAL (- n))) by A4, XBOOLE_0:def 4;
then A11: x in F . n by A1;
n in NAT ;
then n in dom F by FUNCT_2:def 1;
then F . n in rng F by FUNCT_1:def 5;
hence x in union (rng F) by A11, TARSKI:def 4; :: thesis: verum

end;

hence A /\ (great_dom f,-infty ) c= union (rng F) by TARSKI:def 3; :: thesis: verum

end;

union (rng F) c= A /\ (great_dom f,-infty )

proof

for x being set st x in union (rng F) holds
x in A /\ (great_dom f,-infty )

proof

let x be set ; :: thesis: ( x in union (rng F) implies x in A /\ (great_dom f,-infty ) )
assume x in union (rng F) ; :: thesis: x in A /\ (great_dom f,-infty )
then consider Y being set such that
A12: ( x in Y & Y in rng F ) by TARSKI:def 4;
consider m being Element of NAT such that
A13: ( m in dom F & F . m = Y ) by A12, PARTFUN1:26;
x in A /\ (great_dom f,(R_EAL (- m))) by A1, A12, A13;
then A14: ( x in A & x in great_dom f,(R_EAL (- m)) ) by XBOOLE_0:def 4;
then A15: x in dom f by Def14;
A16: R_EAL (- m) < f . x by A14, Def14;
reconsider x = x as Element of X by A12;
-infty < f . x by A16, XXREAL_0:2, XXREAL_0:12;
then x in great_dom f,-infty by A15, Def14;
hence x in A /\ (great_dom f,-infty ) by A14, XBOOLE_0:def 4; :: thesis: verum

end;

hence union (rng F) c= A /\ (great_dom f,-infty ) by TARSKI:def 3; :: thesis: verum

end;

hence A /\ (great_dom f,-infty ) = union (rng F) by A2, XBOOLE_0:def 10; :: thesis: verum