reserve I for set,
  x,x1,x2,y,z for set,
  A for non empty set;
reserve C,D for Category;
reserve a,b,c,d for Object of C;
reserve f,g,h,i,j,k,p1,p2,q1,q2,i1,i2,j1,j2 for Morphism of C;
reserve f for Morphism of a,b,
        g for Morphism of b,a;
reserve g for Morphism of b,c;
reserve f,g for Morphism of C;

theorem
  for F being Injections_family of dom f,I holds (F opp)*(f opp) = (f*F) opp
proof
  let F be Injections_family of dom f, I;
  now
    let x;
    assume
A1: x in I;
    then
A2: cod(F/.x) = (cods F)/.x by Def2
      .= (I --> dom f)/.x by Def16
      .= dom f by A1,Th2;
     reconsider ff=f as Morphism of dom f,cod f by CAT_1:4;
     reconsider gg=F/.x as Morphism of dom(F/.x),dom f by A2,CAT_1:4;
A3:   Hom(dom f,cod f)<>{} & Hom(dom(F/.x),cod(F/.x))<>{} by CAT_1:2;
     then
A4:   ff opp = f opp by OPPCAT_1:def 6;
A5:   gg opp = (F/.x)opp by A3,A2,OPPCAT_1:def 6;
    thus ((F opp)*(f opp))/.x = ((F opp)/.x)(*)(f opp) by A1,Def5
      .= ((F/.x)opp)(*)(f opp) by A1,Def3
      .= (f(*)(F/.x))opp by A2,A4,A5,A3,OPPCAT_1:16
      .= ((f*F)/.x)opp by A1,Def6
      .= ((f*F) opp)/.x by A1,Def3;
  end;
  hence thesis by Th1;
end;
