reserve x,y for set;

theorem
  for A, B being AltCatStr st A, B have_the_same_composition holds
  Intersect(A, B) = Intersect(B, 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 thesis by A1,A2,Def3;
end;
