
theorem Th12:
  for A,B being transitive non empty AltCatStr st A,B are_opposite
  holds A is with_units implies B is with_units
proof
  let A,B be transitive non empty AltCatStr such that
A1: A,B are_opposite;
  assume A is with_units;
  then reconsider A as with_units transitive non empty AltCatStr;
  deffunc C(set,set,set,set,set) = ((the Comp of A).($3,$2,$1)).($4,$5);
A2: now
    let a,b,c be Object of B such that
A3: <^a,b^> <> {} and
A4: <^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;
A5: <^a,b^> = <^b9,a9^> by A1,Th7;
A6: <^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,A3,A4,Th9
      .= C(a,b,c,f,g) by A3,A4,A5,A6,ALTCAT_1:def 8;
  end;
A7: now
    let a be Object of B;
    reconsider a9 = a as Object of A by A1;
    reconsider f = idm a9 as set;
    take f;
    idm a9 in <^a9,a9^>;
    hence f in <^a,a^> by A1,Th7;
    let b be Object of B, g be set;
    reconsider b9 = b as Object of A by A1;
    assume
A8: g in <^a,b^>;
    then
A9: g in <^b9,a9^> by A1,Th7;
    reconsider g9 = g as Morphism of b9,a9 by A1,A8,Th7;
    thus C(a,a,b,f,g) = (idm a9)*g9 by A9,ALTCAT_1:def 8
      .= g by A9,ALTCAT_1:20;
  end;
A10: now
    let a be Object of B;
    reconsider a9 = a as Object of A by A1;
    reconsider f = idm a9 as set;
    take f;
    idm a9 in <^a9,a9^>;
    hence f in <^a,a^> by A1,Th7;
    let b be Object of B, g be set;
    reconsider b9 = b as Object of A by A1;
    assume
A11: g in <^b,a^>;
    then
A12: g in <^a9,b9^> by A1,Th7;
    reconsider g9 = g as Morphism of a9,b9 by A1,A11,Th7;
    thus C(b,a,a,g,f) = g9*(idm a9) by A12,ALTCAT_1:def 8
      .= g by A12,ALTCAT_1:def 17;
  end;
  thus thesis from CatUnitsSch(A2,A7,A10);
end;
