reserve e for set;
reserve C,C1,C2,C3 for AltCatStr;
reserve C for non empty AltCatStr,
  o for Object of C;
reserve C for non empty transitive AltCatStr;

theorem Th25:
  for D being SubCatStr of C st the carrier of D = the carrier of
C & the Arrows of D = the Arrows of C holds the AltCatStr of D = the AltCatStr
  of C
proof
  let D be SubCatStr of C such that
A1: the carrier of D = the carrier of C and
A2: the Arrows of D = the Arrows of C;
A3: D is transitive
  proof
    let o1,o2,o3 be Object of D;
    reconsider p1 = o1, p2 = o2, p3 = o3 as Object of C by A1;
    assume
A4: <^o1,o2^> <> {} & <^o2,o3^> <> {};
A5: <^o1,o3^> = <^p1,p3^> by A2;
    <^o1,o2^> = <^p1,p2^> & <^o2,o3^> = <^p2,p3^> by A2;
    hence thesis by A5,A4,ALTCAT_1:def 2;
  end;
A6: C is SubCatStr of C by Th20;
  D is non empty by A1;
  then C is SubCatStr of D by A1,A2,A3,A6,Th24;
  hence thesis by Th22;
end;
