
theorem Th11:
  for A,B being transitive non empty AltCatStr st A,B are_opposite
  holds A is associative implies B is associative
proof
  let A,B be transitive non empty AltCatStr such that
A1: A,B are_opposite and
A2: A is associative;
  deffunc C(set,set,set,set,set) = ((the Comp of A).($3,$2,$1)).($4,$5);
A3: now
    let a,b,c be Object of B such that
A4: <^a,b^> <> {} and
A5: <^b,c^> <> {};
    let f be Morphism of a,b, g be Morphism of b,c;
    reconsider a9 = a, b9 = b, c9 = c as Object of A by A1;
A6: <^a,b^> = <^b9,a9^> by A1,Th7;
A7: <^b,c^> = <^c9,b9^> by A1,Th7;
    reconsider f9 = f as Morphism of b9, a9 by A1,Th7;
    reconsider g9 = g as Morphism of c9, b9 by A1,Th7;
    thus g*f = f9*g9 by A1,A4,A5,Th9
      .= C(a,b,c,f,g) by A4,A5,A6,A7,ALTCAT_1:def 8;
  end;
A8: now
    let a,b,c,d be Object of B, f,g,h be set;
    reconsider a9 = a, b9 = b, c9 = c, d9 = d as Object of A by A1;
    assume
A9: f in <^a,b^>;
    then
A10: f in <^b9,a9^> by A1,Th9;
    reconsider f9 = f as Morphism of b9,a9 by A1,A9,Th9;
    assume
A11: g in <^b,c^>;
    then
A12: g in <^c9,b9^> by A1,Th9;
    reconsider g9 = g as Morphism of c9,b9 by A1,A11,Th9;
    assume
A13: h in <^c,d^>;
    then
A14: h in <^d9,c9^> by A1,Th9;
    reconsider h9 = h as Morphism of d9,c9 by A1,A13,Th9;
A15: <^c9,a9^> <> {} by A10,A12,ALTCAT_1:def 2;
A16: <^d9,b9^> <> {} by A12,A14,ALTCAT_1:def 2;
    thus C(a,c,d,C(a,b,c,f,g),h) = C(a,c,d,f9*g9,h) by A10,A12,ALTCAT_1:def 8
      .= (f9*g9)*h9 by A14,A15,ALTCAT_1:def 8
      .= f9*(g9*h9) by A2,A10,A12,A14
      .= C(a,b,d,f,g9*h9) by A10,A16,ALTCAT_1:def 8
      .= C(a,b,d,f,C(b,c,d,g,h)) by A12,A14,ALTCAT_1:def 8;
  end;
  thus thesis from CatAssocSch(A3,A8);
end;
