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 Th24:
  a <= b & f is_integrable_on [' a,b '] & g is_integrable_on [' a,
  b '] & f|[' a,b '] is bounded & g|[' a,b '] is bounded & [' a,b '] c= dom f &
[' a,b '] c= dom g & c in ['a,b '] & d in ['a,b '] implies integral(f+g,c,d) =
  integral(f,c,d) + integral(g,c,d) & integral(f-g,c,d) = integral(f,c,d) -
  integral(g,c,d)
proof
  assume
A1: a <= b & f is_integrable_on [' a,b '] & g is_integrable_on [' a,b ']
& f| [' a,b '] is bounded & g|[' a,b '] is bounded & [' a,b '] c= dom f & [' a,
  b '] c= dom g & c in ['a,b '] & d in ['a,b '];
  now
    assume
A2: not c <= d;
    then
A3: integral(f,c,d) = -integral(f,[' d,c ']) by INTEGRA5:def 4;
A4: integral(g,c,d) = -integral(g,[' d,c ']) by A2,INTEGRA5:def 4;
    integral(f+g,c,d) = -integral(f+g,[' d,c ']) by A2,INTEGRA5:def 4;
    hence integral(f+g,c,d) = -integral(f+g,d,c) by A2,INTEGRA5:def 4
      .= -(integral(f,d,c)+integral(g,d,c)) by A1,A2,Lm11
      .= -integral(f,d,c)-integral(g,d,c)
      .= integral(f,c,d) -integral(g,d,c) by A2,A3,INTEGRA5:def 4
      .= integral(f,c,d) +integral(g,c,d) by A2,A4,INTEGRA5:def 4;
    integral(f-g,c,d) = -integral(f-g,[' d,c ']) by A2,INTEGRA5:def 4;
    hence integral(f-g,c,d) =-integral(f-g,d,c) by A2,INTEGRA5:def 4
      .= -(integral(f,d,c)-integral(g,d,c)) by A1,A2,Lm11
      .= -(integral(f,d,c)+integral(g,c,d)) by A2,A4,INTEGRA5:def 4
      .=-integral(f,d,c)-integral(g,c,d)
      .= integral(f,c,d) -integral(g,c,d) by A2,A3,INTEGRA5:def 4;
  end;
  hence thesis by A1,Lm11;
end;
