reserve a,b,x,y,z,z1,z2,z3,y1,y3,y4,A,B,C,D,G,M,N,X,Y,Z,W0,W00 for set,
  R,S,T, W,W1,W2 for Relation,
  F,H,H1 for Function;

theorem Th4:
  for x,y,W st x in field W & y in field W & W is well-ordering
  holds x in W-Seg(y) implies not [y,x] in W
proof
  let x,y,W;
  assume that
A1: x in field W & y in field W and
A2: W is well-ordering;
  W is antisymmetric by A2;
  then
A3: W is_antisymmetric_in field W by RELAT_2:def 12;
  assume x in W-Seg(y);
  then
A4: x<>y & [x,y] in W by WELLORD1:1;
  assume [y,x] in W;
  hence contradiction by A1,A3,A4,RELAT_2:def 4;
end;
