reserve x,y for set;

theorem Th23:
  for A, B being AltCatStr st A, B have_the_same_composition for
  a1,a2 being Object of A, b1,b2 being Object of B for o1,o2 being Object of
Intersect(A, B) st o1 = a1 & o1 = b1 & o2 = a2 & o2 = b2 & <^a1,a2^> <> {} & <^
b1,b2^> <> {} for f being Morphism of a1,a2, g being Morphism of b1,b2 st f = g
  holds f in <^o1,o2^>
proof
  let A, B be AltCatStr such that
A1: A, B have_the_same_composition;
  let a1,a2 be Object of A, b1,b2 be Object of B;
  let o1,o2 be Object of Intersect(A, B);
  assume o1 = a1 & o1 = b1 & o2 = a2 & o2 = b2;
  then <^o1,o2^> = <^a1,a2^> /\ <^b1,b2^> by A1,Th21;
  hence thesis by XBOOLE_0:def 4;
end;
