let X be set ; :: thesis: for S being SigmaField of X
for F being sequence of S
for f being PartFunc of X,ExtREAL
for A being set st ( for n being Nat holds F . n = A /\ (less_dom (f,(- n))) ) holds
A /\ (eq_dom (f,-infty)) = meet (rng F)
let S be SigmaField of X; :: thesis: for F being sequence of S
for f being PartFunc of X,ExtREAL
for A being set st ( for n being Nat holds F . n = A /\ (less_dom (f,(- n))) ) holds
A /\ (eq_dom (f,-infty)) = meet (rng F)
let F be sequence of S; :: thesis: for f being PartFunc of X,ExtREAL
for A being set st ( for n being Nat holds F . n = A /\ (less_dom (f,(- n))) ) holds
A /\ (eq_dom (f,-infty)) = meet (rng F)
let f be PartFunc of X,ExtREAL; :: thesis: for A being set st ( for n being Nat holds F . n = A /\ (less_dom (f,(- n))) ) holds
A /\ (eq_dom (f,-infty)) = meet (rng F)
let A be set ; :: thesis: ( ( for n being Nat holds F . n = A /\ (less_dom (f,(- n))) ) implies A /\ (eq_dom (f,-infty)) = meet (rng F) )
assume A1: for n being Nat holds F . n = A /\ (less_dom (f,(- n))) ; :: thesis: A /\ (eq_dom (f,-infty)) = meet (rng F)
for x being object st x in A /\ (eq_dom (f,-infty)) holds
x in meet (rng F)

proof

let x be object ; :: thesis: ( x in A /\ (eq_dom (f,-infty)) implies x in meet (rng F) )
assume A2: x in A /\ (eq_dom (f,-infty)) ; :: thesis: x in meet (rng F)
then A3: x in A by XBOOLE_0:def 4;
A4: x in eq_dom (f,-infty) by A2, XBOOLE_0:def 4;
for Y being set st Y in rng F holds
x in Y

proof

let Y be set ; :: thesis: ( Y in rng F implies x in Y )
( Y in rng F implies x in Y )

proof

assume Y in rng F ; :: thesis: x in Y
then consider m being Element of NAT such that
m in dom F and
A5: Y = F . m by PARTFUN1:3;
A6: Y = A /\ (less_dom (f,(- m))) by A1, A5;
reconsider x = x as Element of X by A2;
A7: f . x = -infty by A4, Def15;
- m in REAL by XREAL_0:def 1;
then ( x in dom f & not - m <= -infty ) by A4, Def15, XXREAL_0:12;
then x in less_dom (f,(- m)) by A7, Def11;
hence x in Y by A3, A6, XBOOLE_0:def 4; :: thesis: verum

end;

hence ( Y in rng F implies x in Y ) ; :: thesis: verum

end;

hence x in meet (rng F) by SETFAM_1:def 1; :: thesis: verum

end;

then A8: A /\ (eq_dom (f,-infty)) c= meet (rng F) ;
for x being object st x in meet (rng F) holds
x in A /\ (eq_dom (f,-infty))

proof

let x be object ; :: thesis: ( x in meet (rng F) implies x in A /\ (eq_dom (f,-infty)) )
reconsider xx = x as set by TARSKI:1;
assume A9: x in meet (rng F) ; :: thesis: x in A /\ (eq_dom (f,-infty))
A10: for m being Element of NAT holds
( x in A & x in dom f & ex y being R_eal st
( y = f . x & y = -infty ) )

proof

let m be Element of NAT ; :: thesis: ( x in A & x in dom f & ex y being R_eal st
( y = f . x & y = -infty ) )
m in NAT ;
then m in dom F by FUNCT_2:def 1;
then F . m in rng F by FUNCT_1:def 3;
then x in F . m by A9, SETFAM_1:def 1;
then A11: x in A /\ (less_dom (f,(- m))) by A1;
then A12: x in less_dom (f,(- m)) by XBOOLE_0:def 4;
for r being Real holds f . xx < r

proof

let r be Real; :: thesis: f . xx < r
consider n being Nat such that
A13: - n <= r by Th9;
n in NAT by ORDINAL1:def 12;
then n in dom F by FUNCT_2:def 1;
then F . n in rng F by FUNCT_1:def 3;
then x in F . n by A9, SETFAM_1:def 1;
then x in A /\ (less_dom (f,(- n))) by A1;
then x in less_dom (f,(- n)) by XBOOLE_0:def 4;
then f . x < - n by Def11;
hence f . xx < r by A13, XXREAL_0:2; :: thesis: verum

end;

then f . x = -infty by Th13;
hence ( x in A & x in dom f & ex y being R_eal st
( y = f . x & y = -infty ) ) by A11, A12, Def11, XBOOLE_0:def 4; :: thesis: verum

end;

1 in NAT ;
then 1 in dom F by FUNCT_2:def 1;
then F . 1 in rng F by FUNCT_1:def 3;
then x in F . 1 by A9, SETFAM_1:def 1;
then x in A /\ (less_dom (f,(- 1))) by A1;
then reconsider x = x as Element of X ;
x in eq_dom (f,-infty) by A10, Def15;
hence x in A /\ (eq_dom (f,-infty)) by A10, XBOOLE_0:def 4; :: thesis: verum

end;

then meet (rng F) c= A /\ (eq_dom (f,-infty)) ;
hence A /\ (eq_dom (f,-infty)) = meet (rng F) by A8, XBOOLE_0:def 10; :: thesis: verum