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 Th43:
  dom f = a implies y .--> f is Projections_family of a,{y}
proof
  set F = y .--> f;
  assume
A1: dom f = a;
  now
    let x;
    assume
A2: x in {y};
    hence (doms F)/.x = dom(F/.x) by Def1
      .= a by A1,A2,Th2
      .= (y .--> a)/.x by A2,Th2;
  end;
  hence doms F = {y} --> a by Th1;
end;
