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 Function of I,the carrier' of C, G being
Projections_family of a,I st doms F = cods G holds F"*"G is Projections_family
  of a,I
proof
  let F be Function of I,the carrier' of C;
  let G be Projections_family of a,I;
  assume doms F = cods G;
  then doms(F"*"G) = doms G by Th18;
  hence doms(F"*"G) = I --> a by Def13;
end;
