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 )

( 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

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

end;thus ( a in A1 implies a in A2 ) :: thesis: ( a in A2 implies a in A1 )

proof

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