 reserve a,b,c,x for Real;
 reserve C for non empty set;

theorem
  for f, g being PartFunc of REAL, REAL st
    g is continuous non empty &
    f = AffineMap (0,0) | (REAL \ ].a,b.[) &
  dom g = [.a,b.] & g.a = 0 & g.b = 0 holds
    f +* g is continuous
  proof
    let f, g be PartFunc of REAL, REAL;
    assume g is continuous non empty &
    f = AffineMap (0,0) | (REAL \ ].a,b.[) &
    dom g = [.a,b.] & g.a = 0 & g.b = 0; then
    consider h being PartFunc of REAL, REAL such that
A2: h = f +* g &
    for x being Real st x in dom h holds h is_continuous_in x by Kluczyk;
    thus thesis by A2;
  end;
