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
  a1 < a2 implies InitSegm(S,a1) c= InitSegm(S,a2)
proof
  assume
A1: a1 < a2;
  let x be object;
  assume
A2: x in InitSegm(S,a1);
  then x in LowerCone{a1} by XBOOLE_0:def 4;
  then consider a such that
A3: a = x and
A4: for b st b in {a1} holds a < b;
  a1 in {a1} by TARSKI:def 1;
  then a < a1 by A4;
  then a < a2 by A1,Th5;
  then for b holds b in {a2} implies a < b by TARSKI:def 1;
  then
A5: x in LowerCone{a2} by A3;
  x in S by A2,XBOOLE_0:def 4;
  hence thesis by A5,XBOOLE_0:def 4;
end;
