let F1, F2 be Subset of ; :: thesis: ( ( for x being set holds
( x in F1 iff ex f being Function of X,{x1,x2} st
( f = x & card (f " {x1}) = k ) ) ) & ( for x being set holds
( x in F2 iff ex f being Function of X,{x1,x2} st
( f = x & card (f " {x1}) = k ) ) ) implies F1 = F2 )
assume that
A3: for x being set holds
( x in F1 iff ex f being Function of X,{x1,x2} st
( x = f & card (f " {x1}) = k ) ) and
A4: for x being set holds
( x in F2 iff ex f being Function of X,{x1,x2} st
( x = f & card (f " {x1}) = k ) ) ; :: thesis: F1 = F2
for x being set holds
( x in F1 iff x in F2 )

proof

let x be set ; :: thesis: ( x in F1 iff x in F2 )
( x in F1 iff ex f being Function of X,{x1,x2} st
( x = f & card (f " {x1}) = k ) ) by A3;
hence ( x in F1 iff x in F2 ) by A4; :: thesis: verum

end;

hence F1 = F2 by TARSKI:2; :: thesis: verum