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;

theorem
  for F being Function of I,the carrier' of C holds F opp opp = F
proof
  let F be Function of I,the carrier' of C;
  now
    thus the carrier' of C = the carrier' of C opp opp;
    let x;
    assume
A1: x in I;
    hence (F opp opp)/.x = ((F opp)/.x) opp by Def3
      .= (F/.x) opp opp by A1,Def3
      .= F/.x;
  end;
  hence thesis by Th1;
end;
