let X, Y be real-membered set ; :: thesis: ( X c= Y implies Cl X c= Cl Y )
assume A1: X c= Y ; :: thesis: Cl X c= Cl Y
set ClX = { A where A is Subset of REAL : ( X c= A & A is closed ) } ;
Y c= Cl Y by Th58;
then X c= Cl Y by A1;
then Cl Y in { A where A is Subset of REAL : ( X c= A & A is closed ) } ;
hence Cl X c= Cl Y by SETFAM_1:3; :: thesis: verum