
theorem Th2:
  for X,Y be non empty set, f be Function of X,Y, S be Field_Subset of X
    st f is bijective holds .:f.:S is Field_Subset of Y
proof
    let X,Y be non empty set, f be Function of X,Y, S be Field_Subset of X;
    assume
A1: f is bijective; then
    .:f is bijective by Th1; then
A2: dom .:f = bool X & rng .:f = bool Y by FUNCT_2:def 1,def 3;
A3: dom f = X & rng f = Y by A1,FUNCT_2:def 1,def 3;

    reconsider S1 = .:f.:S as Subset-Family of Y;

A4: for A,B being set st A in S1 & B in S1 holds A /\ B in S1
    proof
     let A,B be set;
     assume
A5:  A in S1 & B in S1; then
     consider s be object such that
A6:  s in dom .:f & s in S & A = .:f.s by FUNCT_1:def 6;

     consider t be object such that
A7:  t in dom .:f & t in S & B = .:f.t by FUNCT_1:def 6,A5;

     reconsider s,t as Subset of X by A6,A7;
A8: .:f.s = f.:s & .:f.t = f.:t by A1,Th1;
     (f.:s) /\ (f.:t) = f.:(s/\t) by A1,FUNCT_1:62; then
A9: A /\ B = .:f.(s/\ t) by A1,A6,A7,A8,Th1;
     s/\ t in S by FINSUB_1:def 2,A6,A7;
     hence thesis by A2,A9,FUNCT_1:def 6;
    end;

    for A being Subset of Y st A in S1 holds A` in S1
    proof
     let A being Subset of Y;
     assume A in S1; then
     consider s be object such that
A10: s in dom .:f & s in S & A  = .:f.s by FUNCT_1:def 6;
     reconsider s as Subset of X by A10;
A11: .:f.s = f.:s by A1,Th1;
     f.:(s`) = (f.:X) \ f.:s by A1,FUNCT_1:64; then
     f.:(s`) = Y \ (f.:s) by RELAT_1:113,A3; then
A12: .:f.(s`) = A` by A1,A10,A11,Th1;
     s` in S by A10,PROB_1:def 1;
     hence thesis by A2,A12,FUNCT_1:def 6;
    end; then
    S1 is non empty cap-closed compl-closed by A2,A4,FINSUB_1:def 2;
    hence thesis;
end;
