let C be FormalContext; :: thesis: (ObjectDerivation C) . {} = the carrier' of C
reconsider e = {} as Subset of the carrier of C by XBOOLE_1:2;
set A = { a where a is Attribute of C : for o being Object of C st o in e holds
o is-connected-with a } ;
A1: for x being object st x in { a where a is Attribute of C : for o being Object of C st o in e holds
o is-connected-with a } holds
x in the carrier' of C

proof

let x be object ; :: thesis: ( x in { a where a is Attribute of C : for o being Object of C st o in e holds
o is-connected-with a } implies x in the carrier' of C )
assume x in { a where a is Attribute of C : for o being Object of C st o in e holds
o is-connected-with a } ; :: thesis: x in the carrier' of C
then ex x9 being Attribute of C st
( x9 = x & ( for o being Object of C st o in e holds
o is-connected-with x9 ) ) ;
hence x in the carrier' of C ; :: thesis: verum

end;

A2: for x being object st x in the carrier' of C holds
x in { a where a is Attribute of C : for o being Object of C st o in e holds
o is-connected-with a }

proof

let x be object ; :: thesis: ( x in the carrier' of C implies x in { a where a is Attribute of C : for o being Object of C st o in e holds
o is-connected-with a } )
assume x in the carrier' of C ; :: thesis: x in { a where a is Attribute of C : for o being Object of C st o in e holds
o is-connected-with a }
then reconsider x = x as Attribute of C ;
for o being Object of C st o in e holds
o is-connected-with x ;
hence x in { a where a is Attribute of C : for o being Object of C st o in e holds
o is-connected-with a } ; :: thesis: verum

end;

(ObjectDerivation C) . e = { a where a is Attribute of C : for o being Object of C st o in e holds
o is-connected-with a } by Def2;
hence (ObjectDerivation C) . {} = the carrier' of C by A1, A2, TARSKI:2; :: thesis: verum