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 Th3:
  for x,y,W st x in field W & y in field W & W is well-ordering
  holds not x in W-Seg(y) implies [y,x] in W
proof
  let x,y,W;
  assume that
A1: x in field W and
A2: y in field W and
A3: W is well-ordering;
  W is connected by A3;
  then W is_connected_in field W by RELAT_2:def 14;
  then
A4: x<>y implies [x,y] in W or [y,x] in W by A1,A2,RELAT_2:def 6;
  W is reflexive by A3;
  then
A5: W is_reflexive_in field W by RELAT_2:def 9;
  assume not x in W-Seg(y);
  hence thesis by A1,A5,A4,RELAT_2:def 1,WELLORD1:1;
end;
