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;
