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;

theorem Th3:
  a in [: X1,X2,X3 :] iff ex x1,x2,x3 st a = [x1,x2,x3]
proof
  thus a in [: X1,X2,X3 :] implies ex x1,x2,x3 st a = [x1,x2,x3]
  proof
    assume a in [: X1,X2,X3 :];
    then a in [:[:X1,X2:],X3:] by ZFMISC_1:def 3;
    then consider x12, x3 being object such that
A1: x12 in [:X1,X2:] and
A2: x3 in X3 and
A3: a = [x12,x3] by ZFMISC_1:def 2;
    reconsider x3 as Element of X3 by A2;
    consider x1,x2 being object such that
A4: x1 in X1 and
A5: x2 in X2 and
A6: x12 = [x1,x2] by A1,ZFMISC_1:def 2;
    reconsider x2 as Element of X2 by A5;
    reconsider x1 as Element of X1 by A4;
    a = [x1,x2,x3] by A3,A6;
    hence thesis;
  end;
  given x1,x2,x3 such that
A7: a = [x1,x2,x3];
  a = [[x1,x2],x3] by A7;
  then a in [:[:X1,X2:],X3:];
  hence thesis by ZFMISC_1:def 3;
end;
