let E be RealLinearSpace; :: thesis: for F being binary-image-family of E
for A being non empty binary-image of E holds A (+) (union F) = union { (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 (+) (union F) = union { (A (+) X) where X is binary-image of E : X in F }
let A be non empty binary-image of E; :: thesis: A (+) (union F) = union { (A (+) X) where X is binary-image of E : X in F }
A1: for x being object 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 object ; :: 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
A2: ( x = W (+) A & W in F ) ;
( x = A (+) W & W in F ) by A2;
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
A3: ( x = A (+) W & W in F ) ;
( x = W (+) A & W in F ) by A3;
hence x in { (X (+) A) where X is binary-image of E : X in F } ; :: thesis: verum

end;

thus A (+) (union F) = (union F) (+) A
.= union { (X (+) A) where X is binary-image of E : X in F } by Th12
.= union { (A (+) X) where X is binary-image of E : X in F } by A1, TARSKI:2 ; :: thesis: verum