reserve A for non empty closed_interval Subset of REAL;

theorem F633:
for a,b,c be Real, f,g,h be Function of REAL,REAL st
a <= b & b <= c & f is continuous & g is continuous &
h | [.a,c.] = (f| [.a,b.]) +* (g | [.b,c.]) & f.b = g.b
holds integral( h,['a,c']) = integral(f,['a,b']) + integral(g,['b,c'])
proof
 let a,b,c be Real, f,g,h be Function of REAL,REAL;
 assume that
 A1: a <= b & b <= c and
 A2: f is continuous & g is continuous and
 A3: h | [.a,c.] = (f| [.a,b.]) +* (g | [.b,c.]) and
 A4: f.b = g.b;
 B1: f is_integrable_on ['a,c'] & f | ['a,c'] is bounded by FUZZY_6:25,A2;
 B2: g is_integrable_on ['a,c'] & g | ['a,c'] is bounded by FUZZY_6:25,A2;
 reconsider hh = h as PartFunc of REAL,REAL;
 D1p: REAL = dom h by FUNCT_2:def 1; then
 hh | [.a,c.] is continuous by FUZZY_7:41,A1,A2,A4,A3; then
 hh | ['a,c'] is continuous by XXREAL_0:2,A1,INTEGRA5:def 3; then
 h is_integrable_on ['a,c'] by INTEGRA5:11,D1p;
 hence thesis by FUZZY_6:33,B1,B2,A3,A1,A4;
end;
