 reserve A for non empty Subset of REAL;

theorem
  for s be Real, f,g being Function of REAL,REAL holds
    dom ( ( f | ].-infty,s.] ) +* ( g | [.s,+infty.[ ) ) = REAL &
    dom ( ( f | ].-infty,s.[ ) +* ( g | [.s,+infty.[ ) ) = REAL
proof
 let s be Real;
 let f,g be Function of REAL,REAL;
 set F = (f | ].-infty,s.[) +* (g | [.s,+infty.[);
 set g1 = f| (].-infty,s.[);
 set g2 = g| ([.s,+infty.[);
 D3: -infty < s & s < +infty by XXREAL_0:9,XXREAL_0:12,XREAL_0:def 1;
  dom F = (dom g1) \/ (dom g2) by FUNCT_4:def 1
 .= (].-infty,s.[) \/ (dom g2) by FUNCT_2:def 1
 .= (].-infty,s.[) \/ ([.s,+infty.[) by FUNCT_2:def 1
 .= REAL by XXREAL_1:224,XXREAL_1:173,D3;
 hence thesis by FUZZY_6:35;
end;
