reserve x,y for set;

theorem
  for A, B being para-functional semi-functional category holds A, B
  have_the_same_composition
proof
  let A, B be para-functional semi-functional category;
  now
    let a1,a2,a3 be Object of A such that
A1: <^a1,a2^> <> {} and
A2: <^a2,a3^> <> {};
    let b1,b2,b3 be Object of B such that
A3: <^b1,b2^> <> {} & <^b2,b3^> <> {} and
    b1 = a1 and
    b2 = a2 and
    b3 = a3;
    let f1 be Morphism of a1,a2, g1 be Morphism of b1,b2 such that
A4: g1 = f1;
    reconsider f = f1 as Function of the_carrier_of a1, the_carrier_of a2 by A1
,YELLOW18:34;
    let f2 be Morphism of a2,a3, g2 be Morphism of b2,b3 such that
A5: g2 = f2;
A6: <^b1,b3^> <> {} by A3,ALTCAT_1:def 2;
    reconsider g = f2 as Function of the_carrier_of a2, the_carrier_of a3 by A2
,YELLOW18:34;
    <^a1,a3^> <> {} by A1,A2,ALTCAT_1:def 2;
    hence f2 * f1 = g * f by A1,A2,ALTCAT_1:def 12
      .= g2 * g1 by A3,A4,A5,A6,ALTCAT_1:def 12;
  end;
  hence thesis by Th11;
end;
