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 Th24:
  a in InitSegm(S,b) iff a < b & a in S
proof
  a in InitSegm(S,b) iff a in LowerCone{b} & a in S by XBOOLE_0:def 4;
  hence thesis by Th23;
end;
