let E be RealLinearSpace; :: thesis: for F being binary-image-family of E
for A being non empty binary-image of E holds A (+) (meet F) c= meet { (A (+) X) where X is binary-image of E : X in F }
let F be binary-image-family of E; :: thesis: for A being non empty binary-image of E holds A (+) (meet F) c= meet { (A (+) X) where X is binary-image of E : X in F }
let A be non empty binary-image of E; :: thesis: A (+) (meet F) c= meet { (A (+) X) where X is binary-image of E : X in F }
P1: for x being set holds
( x in { (X (+) A) where X is binary-image of E : X in F } iff x in { (A (+) X) where X is binary-image of E : X in F } )

proof

let x be set ; :: thesis: ( x in { (X (+) A) where X is binary-image of E : X in F } iff x in { (A (+) X) where X is binary-image of E : X in F } )

hereby :: thesis: ( x in { (A (+) X) where X is binary-image of E : X in F } implies x in { (X (+) A) where X is binary-image of E : X in F } )

assume x in { (X (+) A) where X is binary-image of E : X in F } ; :: thesis: x in { (A (+) X) where X is binary-image of E : X in F }
then consider W being binary-image of E such that
X1: ( x = W (+) A & W in F ) ;
( x = A (+) W & W in F ) by X1;
hence x in { (A (+) X) where X is binary-image of E : X in F } ; :: thesis: verum

end;

assume x in { (A (+) X) where X is binary-image of E : X in F } ; :: thesis: x in { (X (+) A) where X is binary-image of E : X in F }
then consider W being binary-image of E such that
X1: ( x = A (+) W & W in F ) ;
( x = W (+) A & W in F ) by X1;
hence x in { (X (+) A) where X is binary-image of E : X in F } ; :: thesis: verum

end;

A (+) (meet F) c= meet { (X (+) A) where X is binary-image of E : X in F } by Th110;
hence A (+) (meet F) c= meet { (A (+) X) where X is binary-image of E : X in F } by P1, TARSKI:1; :: thesis: verum