reserve e for set;
reserve C,C1,C2,C3 for AltCatStr;

theorem
  C1 is SubCatStr of C2 & C2 is SubCatStr of C3 implies C1 is SubCatStr of C3
proof
  assume the carrier of C1 c= the carrier of C2 & the Arrows of C1 cc= the
  Arrows of C2 & the Comp of C1 cc= the Comp of C2 & the carrier of C2 c= the
carrier of C3 & the Arrows of C2 cc= the Arrows of C3 & the Comp of C2 cc= the
  Comp of C3;
  hence the carrier of C1 c= the carrier of C3 & the Arrows of C1 cc= the
  Arrows of C3 & the Comp of C1 cc= the Comp of C3 by Th8;
end;
