reserve X1,X2,X3,X4 for set;

theorem Thm10:
  for X,Y being set,S being cap-closed Subset-Family of X,
      f being Function of X,Y st f is one-to-one
  holds f.:S is cap-closed Subset-Family of Y
  proof
    let X,Y be set,
    S be cap-closed Subset-Family of X,
    f be Function of X,Y;
    assume
A1: f is one-to-one;
    now
      let s1,s2 be set;
      assume
A2:   s1 in f.:S & s2 in f.:S;
      consider c1 be Subset of X such that
A3:   c1 in S and
A4:   s1=f.:c1 by A2,FUNCT_2:def 10;
      consider c2 be Subset of X such that
A5:   c2 in S and
A6:   s2=f.:c2 by A2,FUNCT_2:def 10;
      reconsider f12=f.:(c1/\c2) as Subset of Y;
      c1/\c2 in S by A3,A5,FINSUB_1:def 2;
      then f12 in f.:S by FUNCT_2:def 10;
      hence s1/\s2 in f.:S by A1,A4,A6,FUNCT_1:62;
    end;
    hence thesis by FINSUB_1:def 2;
  end;
