reserve x,y for set;

theorem Th20:
  for A, B being AltCatStr st A, B have_the_same_composition holds
  Intersect(A, B) is SubCatStr of A
proof
  let A,B be AltCatStr;
  set AB = Intersect(A,B);
  assume
A1: A, B have_the_same_composition;
  then
A2: the Comp of AB = Intersect(the Comp of A, the Comp of B) by Def3;
  the carrier of AB = (the carrier of A) /\ (the carrier of B) & the
  Arrows of AB = Intersect(the Arrows of A, the Arrows of B) by A1,Def3;
  hence
  the carrier of AB c= the carrier of A & the Arrows of AB cc= the Arrows
  of A & the Comp of AB cc= the Comp of A by A2,Th15,XBOOLE_1:17;
end;
