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;
