theorem Th45:
  for F being Projections_family of a,I st cod f = a holds F*f is
  Projections_family of dom f,I
proof
  let F be Projections_family of a,I;
  assume cod f = a;
  then doms F = I-->cod f by Def13;
  hence doms(F*f) = I --> dom f by Th16;
end;
