theorem Th31:
  [x, y] in ==>.-relation(TS) implies ex s, v, t, w st x = [s, v] & y = [t, w]
proof
  assume
A1: [x, y] in ==>.-relation(TS);
  then y in [: the carrier of TS, E^omega :] by ZFMISC_1:87;
  then
A2: ex y1, y2 being object
st y1 in the carrier of TS & y2 in E^omega & y = [y1, y2] by
ZFMISC_1:def 2;
  x in [: the carrier of TS, E^omega :] by A1,ZFMISC_1:87;
  then ex x1, x2 being object
st x1 in the carrier of TS & x2 in E^omega & x = [x1, x2] by
ZFMISC_1:def 2;
  hence thesis by A2;
end;
