reserve A for non empty set;
reserve a,b,c,x,y,z for Element of A;
reserve o,o9 for Element of LinPreorders A;
reserve o99 for Element of LinOrders A;

theorem Th5:
  (a <=_o, b or a <_o, b) & (b <=_ o, c or b <_o, c) implies a <=_o, c
proof
  assume a <=_o, b or a <_o, b;
then  a <=_o, b by Th4;
then A1: [a,b] in o;
  assume b <=_o, c or b <_o, c;
then  b <=_o, c by Th4;
then  [b,c] in o;
  hence [a,c] in o by A1,Def1;
end;
