let D1, D2 be set ; :: thesis: ( ( for x being set holds
( x in D1 iff x is FinSequence of D ) ) & ( for x being set holds
( x in D2 iff x is FinSequence of D ) ) implies D1 = D2 )
assume that
A2: for x being set holds
( x in D1 iff x is FinSequence of D ) and
A3: for x being set holds
( x in D2 iff x is FinSequence of D ) ; :: thesis: D1 = D2

now :: thesis: for x being set holds
( ( x in D1 implies x in D2 ) & ( x in D2 implies x in D1 ) )

let x be set ; :: thesis: ( ( x in D1 implies x in D2 ) & ( x in D2 implies x in D1 ) )
thus ( x in D1 implies x in D2 ) :: thesis: ( x in D2 implies x in D1 )

proof

assume x in D1 ; :: thesis: x in D2
then x is FinSequence of D by A2;
hence x in D2 by A3; :: thesis: verum

end;

assume x in D2 ; :: thesis: x in D1
then x is FinSequence of D by A3;
hence x in D1 by A2; :: thesis: verum

end;

hence D1 = D2 by TARSKI:1; :: thesis: verum