
theorem Th26:
  for f, g being non-empty Function
  st dom g = dom f & for i being object st i in dom g holds g.i c= f.i
  holds for i being object st i in dom g holds proj(f,i).:product g = g.i
proof
  let f, g be non-empty Function;
  assume that
    A1: dom g = dom f and
    A2: for i being object st i in dom g holds g.i c= f.i;
  let i be object;
  assume A3: i in dom g;
  thus proj(f,i).:product g = proj(f,i).:product(f +* g) by A1, FUNCT_4:19
    .= g.i by A1, A2, A3, Th25;
end;
