
theorem Th25:
  for f, g being non-empty Function
  st dom g c= 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(f+*g) = g.i
proof
  let f, g be non-empty Function;
  assume that
    A1: dom g c= dom f and
    A2: for i being object st i in dom g holds g.i c= f.i;
  A3: dom(f+*g) = dom f \/ dom g by FUNCT_4:def 1
    .= dom f by A1, XBOOLE_1:12;
  A4: product(f+*g) = product f /\ product(f+*g) by A1, A2, Th23,XBOOLE_1:28;
  let i be object;
  assume A5: i in dom g;
  thus proj(f,i).:product(f+*g) = proj(f+*g,i).:product(f+*g) by A4, Th22
    .= proj(f+*g,i).:dom proj(f+*g,i) by CARD_3:def 16
    .= rng proj(f+*g,i) by RELAT_1:113
    .= (f+*g).i by A1, A3, A5, YELLOW17:3
    .= g.i by A5, FUNCT_4:13;
end;
