reserve X,Y,x,y for set;
reserve A for non empty Poset;
reserve a,a1,a2,a3,b,c for Element of A;
reserve S,T for Subset of A;

theorem Th28:
  S c= T implies InitSegm(S,a) c= InitSegm(T,a)
proof
  assume
A1: S c= T;
  let x be object;
  assume x in InitSegm(S,a);
  then x in S & x in LowerCone{a} by XBOOLE_0:def 4;
  hence thesis by A1,XBOOLE_0:def 4;
end;
