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;
