let X be Subset of REAL; :: thesis: for Y being closed Subset of REAL st X c= Y holds
Cl X c= Y
let Y be closed Subset of REAL; :: thesis: ( X c= Y implies Cl X c= Y )
set ClX = { A where A is Subset of REAL : ( X c= A & A is closed ) } ;
assume X c= Y ; :: thesis: Cl X c= Y
then Y in { A where A is Subset of REAL : ( X c= A & A is closed ) } ;
hence Cl X c= Y by SETFAM_1:3; :: thesis: verum