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 Th41:
  for F being Projections_family of a,I st x in I holds dom(F/.x) = a
proof
  let F be Projections_family of a,I such that
A1: x in I;
  (doms F)/.x = (I --> a)/.x by Def13;
  hence dom(F/.x) = (I --> a)/.x by A1,Def1
    .= a by A1,Th2;
end;
