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

theorem Th22:
  for C1,C2 being AltCatStr st C1 is SubCatStr of C2 & C2 is
  SubCatStr of C1 holds the AltCatStr of C1 = the AltCatStr of C2
proof
  let C1,C2 be AltCatStr;
  assume that
A1: the carrier of C1 c= the carrier of C2 & the Arrows of C1 cc= the
  Arrows of C2 and
A2: the Comp of C1 cc= the Comp of C2 and
A3: the carrier of C2 c= the carrier of C1 & the Arrows of C2 cc= the
  Arrows of C1 and
A4: the Comp of C2 cc= the Comp of C1;
  the carrier of C1 = the carrier of C2 & the Arrows of C1 = the Arrows of
  C2 by A1,A3,Th7,XBOOLE_0:def 10;
  hence thesis by A2,A4,Th7;
end;
