let D1, D2 be set ; :: thesis: ( ( for x being set holds

( x in D1 iff x is one-to-one FinSequence of D ) ) & ( for x being set holds

( x in D2 iff x is one-to-one FinSequence of D ) ) implies D1 = D2 )

assume that

A2: for x being set holds

( x in D1 iff x is one-to-one FinSequence of D ) and

A3: for x being set holds

( x in D2 iff x is one-to-one FinSequence of D ) ; :: thesis: D1 = D2

( x in D1 iff x is one-to-one FinSequence of D ) ) & ( for x being set holds

( x in D2 iff x is one-to-one FinSequence of D ) ) implies D1 = D2 )

assume that

A2: for x being set holds

( x in D1 iff x is one-to-one FinSequence of D ) and

A3: for x being set holds

( x in D2 iff x is one-to-one FinSequence of D ) ; :: thesis: D1 = D2

now :: thesis: for x being object holds

( ( x in D1 implies x in D2 ) & ( x in D2 implies x in D1 ) )

hence
D1 = D2
by TARSKI:2; :: thesis: verum( ( x in D1 implies x in D2 ) & ( x in D2 implies x in D1 ) )

let x be object ; :: 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 )

then x is one-to-one FinSequence of D by A3;

hence x in D1 by A2; :: thesis: verum

end;thus ( x in D1 implies x in D2 ) :: thesis: ( x in D2 implies x in D1 )

proof

assume
x in D2
; :: thesis: x in D1
assume
x in D1
; :: thesis: x in D2

then x is one-to-one FinSequence of D by A2;

hence x in D2 by A3; :: thesis: verum

end;then x is one-to-one FinSequence of D by A2;

hence x in D2 by A3; :: thesis: verum

then x is one-to-one FinSequence of D by A3;

hence x in D1 by A2; :: thesis: verum