reserve A for non empty closed_interval Subset of REAL;

theorem :: FUZZY_6:33
for a,b,c being Real, f,g,h be Function of REAL,REAL st
a <= b & b <= c &
f is_integrable_on ['a,c'] & f | ['a,c'] is bounded &
g is_integrable_on ['a,c'] & g | ['a,c'] is bounded &
h = ( f | ].-infty,b.] ) +* ( g | [.b,+infty.[ ) & 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_integrable_on ['a,c'] & f | ['a,c'] is bounded &
     g is_integrable_on ['a,c'] & g | ['a,c'] is bounded and
 A3: h = ( f | ].-infty,b.] ) +* ( g | [.b,+infty.[ ) and
 A4: f.b = g.b;
 A5: h | [.a,c.] = (f| [.a,b.]) +* (g | [.b,c.]) by A1,FUZZY711,A3;
 b in [.a,c.] by A1; then
 b in ['a,c'] by INTEGRA5:def 3,XXREAL_0:2,A1; then
 h is_integrable_on ['a,c'] by Th39,A2,A3,A4;
 hence thesis by FUZZY_6:33,A1,A2,A4,A5;
end;
