reserve v,x,x1,x2,x3,x4,y,y1,y2,y3,y4,z,z1,z2 for object,
  X,X1,X2,X3,X4,Y,Y1,Y2,Y3,Y4,Y5,
  Z,Z1,Z2,Z3,Z4,Z5 for set;
reserve p for pair object;
reserve R for Relation;
reserve xx1 for Element of X1,
  xx2 for Element of X2,
  xx3 for Element of X3;
reserve xx4 for Element of X4;
reserve A1 for Subset of X1,
  A2 for Subset of X2,
  A3 for Subset of X3,
  A4 for Subset of X4;
reserve x for Element of [:X1,X2,X3:];

theorem
  (for z holds z in Z iff ex x1,x2,x3 st x1 in X1 & x2 in X2 & x3 in X3
  & z = [x1,x2,x3]) implies Z = [: X1,X2,X3 :]
proof
  assume
A1: for z holds z in Z iff ex x1,x2,x3 st x1 in X1 & x2 in X2 & x3 in X3
  & z = [x1,x2,x3];
  now
    let z be object;
    thus z in Z implies z in [:[:X1,X2:],X3:]
    proof
      assume z in Z;
      then consider x1,x2,x3 such that
A2:   x1 in X1 & x2 in X2 and
A3:   x3 in X3 & z = [x1,x2,x3] by A1;
      [x1,x2] in [:X1,X2:] by A2,ZFMISC_1:def 2;
      hence thesis by A3,ZFMISC_1:def 2;
    end;
    assume z in [:[:X1,X2:],X3:];
    then consider x12,x3 being object such that
A4: x12 in [:X1,X2:] and
A5: x3 in X3 and
A6: z = [x12,x3] by ZFMISC_1:def 2;
    consider x1,x2 being object such that
A7: x1 in X1 & x2 in X2 and
A8: x12 = [x1,x2] by A4,ZFMISC_1:def 2;
    z = [x1,x2,x3] by A6,A8;
    hence z in Z by A1,A5,A7;
  end;
  then Z = [:[:X1,X2:],X3:] by TARSKI:2;
  hence thesis by ZFMISC_1:def 3;
end;
