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 Th3:
  for A being transitive RelStr, a1,a2,a3 being Element of A holds
  a1 <= a2 & a2 <= a3 implies a1 <= a3
proof
  let A be transitive RelStr, a1,a2,a3 be Element of A;
  assume that
A1: [a1,a2] in the InternalRel of A and
A2: [a2,a3] in the InternalRel of A;
A3: the InternalRel of A is_transitive_in the carrier of A by Def3;
  a1 in the carrier of A by A1,ZFMISC_1:87;
  hence [a1,a3] in the InternalRel of A by A1,A2,A3;
end;
