let X be non empty set ; :: thesis: for S being SigmaField of X
for A being Element of S
for f being PartFunc of X,REAL
for a being Real holds A /\ (eq_dom (f,a)) = (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a))

let S be SigmaField of X; :: thesis: for A being Element of S
for f being PartFunc of X,REAL
for a being Real holds A /\ (eq_dom (f,a)) = (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a))

let A be Element of S; :: thesis: for f being PartFunc of X,REAL
for a being Real holds A /\ (eq_dom (f,a)) = (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a))

let f be PartFunc of X,REAL; :: thesis: for a being Real holds A /\ (eq_dom (f,a)) = (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a))
let a be Real; :: thesis: A /\ (eq_dom (f,a)) = (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a))
now :: thesis: for x being object st x in (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a)) holds
x in A /\ (eq_dom (f,a))
let x be object ; :: thesis: ( x in (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a)) implies x in A /\ (eq_dom (f,a)) )
assume A1: x in (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a)) ; :: thesis: x in A /\ (eq_dom (f,a))
then A2: x in less_eq_dom (f,a) by XBOOLE_0:def 4;
then A3: x in dom f by MESFUNC6:4;
A4: x in A /\ (great_eq_dom (f,a)) by ;
then x in great_eq_dom (f,a) by XBOOLE_0:def 4;
then A5: ex y1 being Real st
( y1 = f . x & a <= y1 ) by MESFUNC6:6;
ex y2 being Real st
( y2 = f . x & y2 <= a ) by ;
then a = f . x by ;
then A6: x in eq_dom (f,a) by ;
x in A by ;
hence x in A /\ (eq_dom (f,a)) by ; :: thesis: verum
end;
then A7: (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a)) c= A /\ (eq_dom (f,a)) by TARSKI:def 3;
now :: thesis: for x being object st x in A /\ (eq_dom (f,a)) holds
x in (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a))
let x be object ; :: thesis: ( x in A /\ (eq_dom (f,a)) implies x in (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a)) )
assume A8: x in A /\ (eq_dom (f,a)) ; :: thesis: x in (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a))
then A9: x in A by XBOOLE_0:def 4;
A10: x in eq_dom (f,a) by ;
then A11: ex y being Real st
( y = f . x & a = y ) by MESFUNC6:7;
A12: x in dom f by ;
then x in great_eq_dom (f,a) by ;
then A13: x in A /\ (great_eq_dom (f,a)) by ;
x in less_eq_dom (f,a) by ;
hence x in (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a)) by ; :: thesis: verum
end;
then A /\ (eq_dom (f,a)) c= (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a)) by TARSKI:def 3;
hence A /\ (eq_dom (f,a)) = (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a)) by ; :: thesis: verum