let A1, A2 be Subset of REAL; :: thesis: ( ( for a being Real holds
( a in A1 iff ex n being Nat st a in dyadic n ) ) & ( for a being Real holds
( a in A2 iff ex n being Nat st a in dyadic n ) ) implies A1 = A2 )
assume that
A2: for x being Real holds
( x in A1 iff ex n being Nat st x in dyadic n ) and
A3: for x being Real holds
( x in A2 iff ex n being Nat st x in dyadic n ) ; :: thesis: A1 = A2
for a being object holds
( a in A1 iff a in A2 )

proof

let a be object ; :: thesis: ( a in A1 iff a in A2 )
thus ( a in A1 implies a in A2 ) :: thesis: ( a in A2 implies a in A1 )

proof

assume A4: a in A1 ; :: thesis: a in A2
then reconsider a = a as Real ;
ex n being Nat st a in dyadic n by A2, A4;
hence a in A2 by A3; :: thesis: verum

end;

thus ( a in A2 implies a in A1 ) :: thesis: verum

proof

assume A5: a in A2 ; :: thesis: a in A1
then reconsider a = a as Real ;
ex n being Nat st a in dyadic n by A3, A5;
hence a in A1 by A2; :: thesis: verum

end;

end;

hence A1 = A2 by TARSKI:2; :: thesis: verum