let E be RealLinearSpace; :: thesis: for F being binary-image-family of E

for B being non empty binary-image of E holds (dilation B) . (union F) = union { ((dilation B) . X) where X is binary-image of E : X in F }

let F be binary-image-family of E; :: thesis: for B being non empty binary-image of E holds (dilation B) . (union F) = union { ((dilation B) . X) where X is binary-image of E : X in F }

let B be non empty binary-image of E; :: thesis: (dilation B) . (union F) = union { ((dilation B) . X) where X is binary-image of E : X in F }

A1: for x being object holds

( x in { (X (+) B) where X is binary-image of E : X in F } iff x in { ((dilation B) . X) where X is binary-image of E : X in F } )

.= union { (X (+) B) where X is binary-image of E : X in F } by Th12

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

for B being non empty binary-image of E holds (dilation B) . (union F) = union { ((dilation B) . X) where X is binary-image of E : X in F }

let F be binary-image-family of E; :: thesis: for B being non empty binary-image of E holds (dilation B) . (union F) = union { ((dilation B) . X) where X is binary-image of E : X in F }

let B be non empty binary-image of E; :: thesis: (dilation B) . (union F) = union { ((dilation B) . X) where X is binary-image of E : X in F }

A1: for x being object holds

( x in { (X (+) B) where X is binary-image of E : X in F } iff x in { ((dilation B) . X) where X is binary-image of E : X in F } )

proof

thus (dilation B) . (union F) =
(union F) (+) B
by Def2
let x be object ; :: thesis: ( x in { (X (+) B) where X is binary-image of E : X in F } iff x in { ((dilation B) . X) where X is binary-image of E : X in F } )

then consider X being binary-image of E such that

A3: ( x = (dilation B) . X & X in F ) ;

( x = X (+) B & X in F ) by A3, Def2;

hence x in { (W (+) B) where W is binary-image of E : W in F } ; :: thesis: verum

end;hereby :: thesis: ( x in { ((dilation B) . X) where X is binary-image of E : X in F } implies x in { (X (+) B) where X is binary-image of E : X in F } )

assume
x in { ((dilation B) . X) where X is binary-image of E : X in F }
; :: thesis: x in { (X (+) B) where X is binary-image of E : X in F } assume
x in { (X (+) B) where X is binary-image of E : X in F }
; :: thesis: x in { ((dilation B) . W) where W is binary-image of E : W in F }

then consider X being binary-image of E such that

A2: ( x = X (+) B & X in F ) ;

( x = (dilation B) . X & X in F ) by A2, Def2;

hence x in { ((dilation B) . W) where W is binary-image of E : W in F } ; :: thesis: verum

end;then consider X being binary-image of E such that

A2: ( x = X (+) B & X in F ) ;

( x = (dilation B) . X & X in F ) by A2, Def2;

hence x in { ((dilation B) . W) where W is binary-image of E : W in F } ; :: thesis: verum

then consider X being binary-image of E such that

A3: ( x = (dilation B) . X & X in F ) ;

( x = X (+) B & X in F ) by A3, Def2;

hence x in { (W (+) B) where W is binary-image of E : W in F } ; :: thesis: verum

.= union { (X (+) B) where X is binary-image of E : X in F } by Th12

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