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,B being set st A is finite & A c= Y & B is finite & B c= Z &
  X c= [:A,B:]
proof
  deffunc G(object) = $1`2;
  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;
  consider g being Function such that
A5: dom g = X and
A6: for a being object st a in X holds g.a = G(a) from FUNCT_1:sch 3;
  take A = rng f, B = rng g;
  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
A7: x in dom f and
A8: f.x = a by FUNCT_1:def 3;
    consider y,z being object such that
A9: y in Y and z in Z and
A10: x = [y,z] by A2,A3,A7,ZFMISC_1:def 2;
    f.x = x`1 by A3,A4,A7
      .= y by A10;
    hence thesis by A8,A9;
  end;
  thus B is finite by A1,A5,Th8;
  thus B c= Z
  proof
    let a be object;
    assume a in B;
    then consider x being object such that
A11: x in dom g and
A12: g.x = a by FUNCT_1:def 3;
    consider y,z being object such that
    y in Y and
A13: z in Z and
A14: x = [y,z] by A2,A5,A11,ZFMISC_1:def 2;
    g.x = x`2 by A5,A6,A11
      .= z by A14;
    hence thesis by A12,A13;
  end;
  thus X c= [:A,B:]
  proof
    let a be object;
    assume
A15: a in X;
    then consider x,y being object such that
    x in Y and y in Z and
A16: a=[x,y] by A2,ZFMISC_1:def 2;
A17: g.a = a`2 by A6,A15
      .= y by A16;
    f.a = a`1 by A4,A15
      .= x by A16;
    then
A18: x in A by A3,A15,FUNCT_1:def 3;
    y in B by A5,A15,A17,FUNCT_1:def 3;
    hence thesis by A16,A18,ZFMISC_1:87;
  end;
end;
