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 Projections_family of cod f,I holds (f opp)*(F opp) = (F*f ) opp
proof
  let F be Projections_family of cod f, I;
  now
    let x;
    assume
A1: x in I;
    then
A2: dom(F/.x) = (doms F)/.x by Def1
      .= (I --> cod f)/.x by Def13
      .= cod 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 cod f,cod(F/.x) 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)(*)((F opp)/.x) by A1,Def6
      .= (f opp)(*)((F/.x)opp) by A1,Def3
      .= (gg(*)ff)opp by A2,A4,A5,A3,OPPCAT_1:16
      .= ((F*f)/.x)opp by A1,Def5
      .= ((F*f) opp)/.x by A1,Def3;
  end;
  hence thesis by Th1;
end;
