theorem Th54:
  z in [: X1,X2,X3 :] implies ex x1,x2,x3 st x1 in X1 & x2 in X2 &
  x3 in X3 & z = [x1,x2,x3]
proof
  assume z in [: X1,X2,X3 :];
  then z 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: z = [x12,x3] by ZFMISC_1:def 2;
  consider x1,x2 being object such that
A4: x1 in X1 & x2 in X2 and
A5: x12 = [x1,x2] by A1,ZFMISC_1:def 2;
  z = [x1,x2,x3] by A3,A5;
  hence thesis by A2,A4;
end;
