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;
