theorem Th11:
  (for a holds a in D iff ex x1,x2,x3,x4 st a = [x1,x2,x3,x4])
  implies D = [: X1,X2,X3,X4 :]
proof
  assume
A1: for a holds a in D iff ex x1,x2,x3,x4 st a = [x1,x2,x3,x4];
  now
    let a be object;
    thus a in D implies a in [:[:X1,X2,X3:],X4:]
    proof
      assume a in D;
      then consider x1,x2,x3,x4 such that
A2:   a = [x1,x2,x3,x4] by A1;
      a = [[x1,x2,x3],x4] by A2;
      hence thesis;
    end;
    assume a in [:[:X1,X2,X3:],X4:];
    then consider x123,x4 being object such that
A3: x123 in [:X1,X2,X3:] and
A4: x4 in X4 and
A5: a = [x123,x4] by ZFMISC_1:def 2;
    reconsider x4 as Element of X4 by A4;
    consider x1,x2,x3 such that
A6: x123 = [x1,x2,x3] by A3,Th3;
    a = [x1,x2,x3,x4] by A5,A6;
    hence a in D by A1;
  end;
  then D = [:[:X1,X2,X3:],X4:] by TARSKI:2;
  hence thesis by ZFMISC_1:def 4;
end;
