reserve i,j,k,x for object;

theorem
  for C being transitive non empty AltCatStr, a1,a2,a3 being Object of C
holds dom((the Comp of C).(a1,a2,a3)) = [:<^a2,a3^>,<^a1,a2^>:] & rng((the Comp
  of C).(a1,a2,a3)) c= <^a1,a3^>
proof
  let C be transitive non empty AltCatStr, a1,a2,a3 be Object of C;
  <^a1,a3^> = {} implies <^a1,a2^> = {} or <^a2,a3^> = {} by Def2;
  then <^a1,a3^> = {} implies [:<^a2,a3^>,<^a1,a2^>:] = {};
  hence thesis by FUNCT_2:def 1,RELAT_1:def 19;
end;
