let C be FormalContext; :: thesis: for o being Object of C holds (ObjectDerivation C) . {o} = { a where a is Attribute of C : o is-connected-with a }
let o be Object of C; :: thesis: (ObjectDerivation C) . {o} = { a where a is Attribute of C : o is-connected-with a }
{o} c= the carrier of C

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in {o} or x in the carrier of C )
assume x in {o} ; :: thesis: x in the carrier of C
then x = o by TARSKI:def 1;
hence x in the carrier of C ; :: thesis: verum

end;

then reconsider t = {o} as Subset of the carrier of C ;
A1: (ObjectDerivation C) . t = { a where a is Attribute of C : for o' being Object of C st o' in t holds
o' is-connected-with a } by Def6;
A2: for x being set st x in { a where a is Attribute of C : for o' being Object of C st o' in t holds
o' is-connected-with a } holds
x in { a where a is Attribute of C : o is-connected-with a }

proof

let x be set ; :: thesis: ( x in { a where a is Attribute of C : for o' being Object of C st o' in t holds
o' is-connected-with a } implies x in { a where a is Attribute of C : o is-connected-with a } )
assume x in { a where a is Attribute of C : for o' being Object of C st o' in t holds
o' is-connected-with a } ; :: thesis: x in { a where a is Attribute of C : o is-connected-with a }
then consider x' being Attribute of C such that
A3: ( x' = x & ( for o' being Object of C st o' in t holds
o' is-connected-with x' ) ) ;
reconsider x = x as Attribute of C by A3;
consider o' being Element of t;
reconsider o' = o' as Object of C by TARSKI:def 1;
A4: o' is-connected-with x by A3;
o' = o by TARSKI:def 1;
hence x in { a where a is Attribute of C : o is-connected-with a } by A4; :: thesis: verum

end;

for x being set st x in { a where a is Attribute of C : o is-connected-with a } holds
x in { a where a is Attribute of C : for o' being Object of C st o' in t holds
o' is-connected-with a }

proof

let x be set ; :: thesis: ( x in { a where a is Attribute of C : o is-connected-with a } implies x in { a where a is Attribute of C : for o' being Object of C st o' in t holds
o' is-connected-with a } )
assume x in { a where a is Attribute of C : o is-connected-with a } ; :: thesis: x in { a where a is Attribute of C : for o' being Object of C st o' in t holds
o' is-connected-with a }
then consider x' being Attribute of C such that
A5: ( x' = x & o is-connected-with x' ) ;
reconsider x = x as Attribute of C by A5;
for o' being Object of C st o' in t holds
o' is-connected-with x by A5, TARSKI:def 1;
hence x in { a where a is Attribute of C : for o' being Object of C st o' in t holds
o' is-connected-with a } ; :: thesis: verum

end;

hence (ObjectDerivation C) . {o} = { a where a is Attribute of C : o is-connected-with a } by A1, A2, TARSKI:2; :: thesis: verum