reserve a,b,c,d for set,
  D,X1,X2,X3,X4 for non empty set,
  x1,y1,z1 for Element of X1,
  x2 for Element of X2,
  x3 for Element of X3,
  x4 for Element of X4,
  A1,B1 for Subset of X1;
reserve x,y for Element of [:X1,X2,X3:];

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;
