let X be set ; :: thesis: for f being PartFunc of X,ExtREAL
for S being SigmaField of X
for F being Function of NAT ,S
for A being set
for r being real number st ( for n being Element of NAT holds F . n = A /\ (less_dom f,(R_EAL (r + (1 / (n + 1))))) ) holds
A /\ (less_eq_dom f,(R_EAL r)) = meet (rng F)
let f be PartFunc of X,ExtREAL ; :: thesis: for S being SigmaField of X
for F being Function of NAT ,S
for A being set
for r being real number st ( for n being Element of NAT holds F . n = A /\ (less_dom f,(R_EAL (r + (1 / (n + 1))))) ) holds
A /\ (less_eq_dom f,(R_EAL r)) = meet (rng F)
let S be SigmaField of X; :: thesis: for F being Function of NAT ,S
for A being set
for r being real number st ( for n being Element of NAT holds F . n = A /\ (less_dom f,(R_EAL (r + (1 / (n + 1))))) ) holds
A /\ (less_eq_dom f,(R_EAL r)) = meet (rng F)
let F be Function of NAT ,S; :: thesis: for A being set
for r being real number st ( for n being Element of NAT holds F . n = A /\ (less_dom f,(R_EAL (r + (1 / (n + 1))))) ) holds
A /\ (less_eq_dom f,(R_EAL r)) = meet (rng F)
let A be set ; :: thesis: for r being real number st ( for n being Element of NAT holds F . n = A /\ (less_dom f,(R_EAL (r + (1 / (n + 1))))) ) holds
A /\ (less_eq_dom f,(R_EAL r)) = meet (rng F)
let r be real number ; :: thesis: ( ( for n being Element of NAT holds F . n = A /\ (less_dom f,(R_EAL (r + (1 / (n + 1))))) ) implies A /\ (less_eq_dom f,(R_EAL r)) = meet (rng F) )
assume A1: for n being Element of NAT holds F . n = A /\ (less_dom f,(R_EAL (r + (1 / (n + 1))))) ; :: thesis: A /\ (less_eq_dom f,(R_EAL r)) = meet (rng F)
A2: for x being set st x in A /\ (less_eq_dom f,(R_EAL r)) holds
x in meet (rng F)

proof

let x be set ; :: thesis: ( x in A /\ (less_eq_dom f,(R_EAL r)) implies x in meet (rng F) )
assume A3: x in A /\ (less_eq_dom f,(R_EAL r)) ; :: thesis: x in meet (rng F)
A4: x in A by A3, XBOOLE_0:def 4;
A5: x in less_eq_dom f,(R_EAL r) by A3, XBOOLE_0:def 4;
A6: 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 )
A7: ( Y in rng F implies x in Y )

proof

assume A8: Y in rng F ; :: thesis: x in Y
consider m being Element of NAT such that
m in dom F and
A9: Y = F . m by A8, PARTFUN1:26;
A10: Y = A /\ (less_dom f,(R_EAL (r + (1 / (m + 1))))) by A1, A9;
A11: x in dom f by A5, Def13;
reconsider x = x as Element of X by A3;
A12: f . x <= R_EAL r by A5, Def13;
A13: (m + 1) " > 0 ;
A14: 1 / (m + 1) > 0 by A13, XCMPLX_1:217;
A15: R_EAL r < R_EAL (r + (1 / (m + 1))) by A14, XREAL_1:31;
A16: f . x < R_EAL (r + (1 / (m + 1))) by A12, A15, XXREAL_0:2;
A17: x in less_dom f,(R_EAL (r + (1 / (m + 1)))) by A11, A16, Def12;
thus x in Y by A4, A10, A17, XBOOLE_0:def 4; :: thesis: verum

end;

thus ( Y in rng F implies x in Y ) by A7; :: thesis: verum

end;

thus x in meet (rng F) by A6, SETFAM_1:def 1; :: thesis: verum

end;

A18: A /\ (less_eq_dom f,(R_EAL r)) c= meet (rng F) by A2, TARSKI:def 3;
A19: for x being set st x in meet (rng F) holds
x in A /\ (less_eq_dom f,(R_EAL r))

proof

let x be set ; :: thesis: ( x in meet (rng F) implies x in A /\ (less_eq_dom f,(R_EAL r)) )
assume A20: x in meet (rng F) ; :: thesis: x in A /\ (less_eq_dom f,(R_EAL r))
A21: for m being Element of NAT holds
( x in A & x in dom f & f . x < R_EAL (r + (1 / (m + 1))) )

proof

let m be Element of NAT ; :: thesis: ( x in A & x in dom f & f . x < R_EAL (r + (1 / (m + 1))) )
A22: m in NAT ;
A23: m in dom F by A22, FUNCT_2:def 1;
A24: F . m in rng F by A23, FUNCT_1:def 5;
A25: x in F . m by A20, A24, SETFAM_1:def 1;
A26: x in A /\ (less_dom f,(R_EAL (r + (1 / (m + 1))))) by A1, A25;
A27: x in less_dom f,(R_EAL (r + (1 / (m + 1)))) by A26, XBOOLE_0:def 4;
thus ( x in A & x in dom f & f . x < R_EAL (r + (1 / (m + 1))) ) by A26, A27, Def12, XBOOLE_0:def 4; :: thesis: verum

end;

reconsider y = f . x as R_eal ;
A28: 1 in NAT ;
A29: 1 in dom F by A28, FUNCT_2:def 1;
A30: F . 1 in rng F by A29, FUNCT_1:def 5;
A31: x in F . 1 by A20, A30, SETFAM_1:def 1;
A32: x in A /\ (less_dom f,(R_EAL (r + (1 / (1 + 1))))) by A1, A31;
reconsider x = x as Element of X by A32;
A33: y <= R_EAL r

proof

A34: now

per cases ( y = -infty or not y = -infty ) ;

suppose A35: y = -infty ; :: thesis: y <= R_EAL r

thus y <= R_EAL r by A35, XXREAL_0:5; :: thesis: verum

end;

suppose A36: not y = -infty ; :: thesis: y <= R_EAL r

A37: not y <= -infty by A36, XXREAL_0:6;
A38: y < R_EAL (r + (1 / (1 + 1))) by A21;
reconsider y1 = y as Real by A37, A38, XXREAL_0:48;
A39: for m being Element of NAT holds y1 - (1 / (m + 1)) <= r

proof

let m be Element of NAT ; :: thesis: y1 - (1 / (m + 1)) <= r
A40: R_EAL y1 < R_EAL (r + (1 / (m + 1))) by A21;
thus y1 - (1 / (m + 1)) <= r by A40, XREAL_1:22; :: thesis: verum

end;

thus y <= R_EAL r by A39, Th14; :: thesis: verum

end;

end;

end;

thus y <= R_EAL r by A34; :: thesis: verum

end;

A41: x in less_eq_dom f,(R_EAL r) by A21, A33, Def13;
thus x in A /\ (less_eq_dom f,(R_EAL r)) by A21, A41, XBOOLE_0:def 4; :: thesis: verum

end;

A42: meet (rng F) c= A /\ (less_eq_dom f,(R_EAL r)) by A19, TARSKI:def 3;
thus A /\ (less_eq_dom f,(R_EAL r)) = meet (rng F) by A18, A42, XBOOLE_0:def 10; :: thesis: verum