reserve a,b,c,d,e,x,r for Real,
  A for non empty closed_interval Subset of REAL,
  f,g for PartFunc of REAL,REAL;

theorem Th12:
  a<=b & [' a,b '] c= dom f & [' a,b '] c= dom g & f
  is_integrable_on [' a,b '] & g is_integrable_on [' a,b '] & f|[' a,b '] is
  bounded & g|[' a,b '] is bounded implies integral(f+g,a,b) =integral(f,a,b) +
  integral(g,a,b) & integral(f-g,a,b) =integral(f,a,b) - integral(g,a,b)
proof
  assume that
A1: a<=b and
A2: [' a,b '] c= dom f & [' a,b '] c= dom g & f is_integrable_on [' a,b
  '] & g is_integrable_on [' a,b '] & f|[' a,b '] is bounded & g|[' a,b '] is
  bounded;
A3: integral(f+g,a,b) = integral(f+g,[' a,b ']) & integral(f-g,a,b) =
  integral(f -g,[' a,b ']) by A1,INTEGRA5:def 4;
  integral(f,a,b) = integral(f,[' a,b ']) & integral(g,a,b) = integral(g,
  [' a, b ']) by A1,INTEGRA5:def 4;
  hence thesis by A2,A3,Th11;
end;
