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 Th32:
  a1 < a2 & a1 in S & a2 in T & T is Initial_Segm of S implies a1 in T
proof
  assume that
A1: a1 < a2 and
A2: a1 in S and
A3: a2 in T and
A4: T is Initial_Segm of S;
  consider a such that
  a in S and
A5: T = InitSegm(S,a) by A2,A4,Def11;
  now
    let b;
    assume b in {a};
    then
A6: b = a by TARSKI:def 1;
    a2 in LowerCone{a} by A3,A5,XBOOLE_0:def 4;
    then
A7: ex a3 st a3 = a2 & for c st c in {a} holds a3 < c;
    a in {a} by TARSKI:def 1;
    then a2 < a by A7;
    hence a1 < b by A1,A6,Th5;
  end;
  then a1 in LowerCone{a};
  hence thesis by A2,A5,XBOOLE_0:def 4;
end;
