reserve A, B, X, Y, Z, x, y for set;
reserve f for Function;
reserve O for Ordinal;
reserve a for set;

theorem
  X is finite & X c= [:Y,Z:] implies
  ex A being set st A is finite & A c= Y & X c= [:A,Z:]
proof
  assume that
A1: X is finite and
A2: X c= [:Y,Z:];
  consider f being Function such that
A3: dom f = X and
A4: for a being object st a in X holds f.a = F(a) from FUNCT_1:sch 3;
  take A = rng f;
  thus A is finite by A1,A3,Th8;
  thus A c= Y
  proof
    let a be object;
    assume a in A;
    then consider x being object such that
A5: x in dom f and
A6: f.x = a by FUNCT_1:def 3;
    consider y,z being object such that
A7: y in Y and z in Z and
A8: x = [y,z] by A2,A3,A5,ZFMISC_1:def 2;
    f.x = x`1 by A3,A4,A5
      .= y by A8;
    hence thesis by A6,A7;
  end;
  thus X c= [:A,Z:]
  proof
    let a be object;
    assume
A9: a in X;
    then consider x,y being object such that
    x in Y and
A10: y in Z and
A11: a=[x,y] by A2,ZFMISC_1:def 2;
    f.a = a`1 by A4,A9
      .= x by A11;
    then x in A by A3,A9,FUNCT_1:def 3;
    hence thesis by A10,A11,ZFMISC_1:87;
  end;
end;
