theorem Th10:
  a in [: X1,X2,X3,X4 :] iff ex x1,x2,x3,x4 st a = [x1,x2,x3,x4]
proof
  thus a in [: X1,X2,X3,X4 :] implies ex x1,x2,x3,x4 st a = [x1,x2,x3,x4]
  proof
    assume a in [: X1,X2,X3,X4 :];
    then a in [:[:X1,X2,X3:],X4:] by ZFMISC_1:def 4;
    then consider x123, x4 being object such that
A1: x123 in [:X1,X2,X3:] and
A2: x4 in X4 and
A3: a = [x123,x4] by ZFMISC_1:def 2;
    reconsider x4 as Element of X4 by A2;
    consider x1 being (Element of X1), x2 being (Element of X2), x3 being
    Element of X3 such that
A4: x123 = [x1,x2,x3] by A1,Th3;
    a = [x1,x2,x3,x4] by A3,A4;
    hence thesis;
  end;
  given x1,x2,x3,x4 such that
A5: a = [x1,x2,x3,x4];
  a = [[x1,x2,x3],x4] by A5;
  then a in [:[:X1,X2,X3:],X4:];
  hence thesis by ZFMISC_1:def 4;
end;
