reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;
reserve X,X1,X2 for Subset of A;
reserve Y for Subset of B;
reserve R,R1,R2 for Subset of [:A,B:];
reserve FR for Subset-Family of [:A,B:];

theorem :: (7.2.1)
  for G being Subset-Family of B st
  G = {R.:^X where R is Subset of [:A,B:]: R in FR} holds
  (Intersect FR).:^X = Intersect G
proof
  let G be Subset-Family of B;
  assume
A1: G = {R.:^X where R is Subset of [:A,B:]: R in FR};
A2: for x being set st G = {x} holds Intersect G = x
  proof
    let x be set;
    assume G = {x};
    then Intersect G = meet {x} by SETFAM_1:def 9;
    hence thesis by SETFAM_1:10;
  end;
  per cases;
  suppose
A3: X = {};
    then
A4: (Intersect FR).:^X = B by Th29;
    G c= {B}
    proof
      let a be object;
      assume a in G;
      then ex R being Subset of [:A,B:] st ( a = R.:^X)&( R in FR) by A1;
      then a = B by A3,Th29;
      hence thesis by TARSKI:def 1;
    end;
    then G = {} or G = {B} by ZFMISC_1:33;
    hence thesis by A2,A4,SETFAM_1:def 9;
  end;
  suppose
A5: X <> {};
    per cases;
    suppose
A6:   FR = {};
      then Intersect FR = [:A,B:] by SETFAM_1:def 9;
      then
A7:   (Intersect FR).:^X = B by Th37;
      G c= {B}
      proof
        let a be object;
        assume a in G;
        then ex R being Subset of [:A,B:] st ( a = R.:^X)&( R in FR) by A1;
        hence thesis by A6;
      end;
      then G = {} or G = {B} by ZFMISC_1:33;
      hence thesis by A2,A7,SETFAM_1:def 9;
    end;
    suppose
A8:   FR <> {};
      per cases;
      suppose
A9:     B = {};
        then [:A,B:] = {} by ZFMISC_1:90;
        then FR = {} or FR = {{}} by ZFMISC_1:1,33;
        then Intersect FR = meet {{}} by A8,SETFAM_1:def 9;
        then (Intersect FR).:^X = {} by A5,Th36,SETFAM_1:10;
        hence thesis by A9;
      end;
      suppose B <> {};
        then reconsider B as non empty set;
        thus (Intersect FR).:^X c= Intersect G
        proof
          let a be object;
          assume
A10:      a in (Intersect FR).:^X;
          then
A11:      a in B;
          reconsider a as Element of B by A10;
          G <> {} implies a in Intersect G
          proof
            assume
A12:        G <> {};
            then
A13:        Intersect G = meet G by SETFAM_1:def 9;
            for Y being set holds Y in G implies a in Y
            proof
              let Y be set;
              assume Y in G;
              then consider R being Subset of [:A,B:] such that
A14:          Y = R.:^X and
A15:          R in FR by A1;
              for x being set st x in X holds a in Im(R,x)
              proof
                let x be set;
                assume x in X;
                then a in Im(Intersect FR,x) by A10,Th24;
                then [x,a] in Intersect FR by Th9;
                then [x,a] in meet FR by A8,SETFAM_1:def 9;
                then [x,a] in R by A15,SETFAM_1:def 1;
                hence thesis by Th9;
              end;
              hence thesis by A14,Th25;
            end;
            hence thesis by A12,A13,SETFAM_1:def 1;
          end;
          hence thesis by A11,SETFAM_1:def 9;
        end;
        let a be object;
        assume
A16:    a in Intersect G;
        then reconsider a as Element of B;
        consider R being Subset of [:A,B:] such that
A17:    R in FR by A8,SUBSET_1:4;
        R.:^X in G by A1,A17;
        then
A18:    a in meet G by A16,SETFAM_1:def 9;
        for x being set st x in X holds a in Im(Intersect FR,x)
        proof
          let x be set such that
A19:      x in X;
          for Y being set holds Y in FR implies [x,a] in Y
          proof
            let P be set;
            assume
A20:        P in FR;
            then reconsider P as Subset of [:A,B:];
            set S = P.:^X;
            S in G by A1,A20;
            then a in P.:^X by A18,SETFAM_1:def 1;
            then a in Im(P,x) by A19,Th24;
            hence thesis by Th9;
          end;
          then [x,a] in meet FR by A8,SETFAM_1:def 1;
          then [x,a] in Intersect FR by A8,SETFAM_1:def 9;
          hence thesis by Th9;
        end;
        hence thesis by Th25;
      end;
    end;
  end;
end;
