let X1, X2 be Subset of ({0,1} ^omega); :: thesis: ( ( for x being set holds
( x in X1 iff ex p being XFinSequence of NAT st
( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ) ) & ( for x being set holds
( x in X2 iff ex p being XFinSequence of NAT st
( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ) ) implies X1 = X2 )
assume that
A5: for x being set holds
( x in X1 iff ex p being XFinSequence of NAT st
( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ) and
A6: for x being set holds
( x in X2 iff ex p being XFinSequence of NAT st
( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ) ; :: thesis: X1 = X2
for x being object holds
( x in X1 iff x in X2 )

proof

let x be object ; :: thesis: ( x in X1 iff x in X2 )
( x in X1 iff ex p being XFinSequence of NAT st
( p = x & p is dominated_by_0 & dom p = n & Sum p = m ) ) by A5;
hence ( x in X1 iff x in X2 ) by A6; :: thesis: verum

end;

hence X1 = X2 by TARSKI:2; :: thesis: verum