reserve A for non empty closed_interval Subset of REAL;

theorem
for a,b,c being Real, f,g,F be Function of REAL,REAL st
a <= b & b <= c &
F = f | [.a,b.] +* g | [.b,c.]
holds
F is Function of ['a,c'],REAL
proof
 let a,b,c be Real;
 let f,g,F be Function of REAL,REAL;
 set g1 = f | [.a,b.];
 set g2 = g | [.b,c.];
 assume that
 A1: a <= b & b <= c and
 A2: F = f | [.a,b.] +* g | [.b,c.];
 Dg: dom F = (dom g1) \/ (dom g2) by FUNCT_4:def 1,A2
 .= ([.a,b.]) \/ (dom g2) by FUNCT_2:def 1
 .=([.a,b.]) \/ ( [.b,c.]) by FUNCT_2:def 1
 .=([. a,c .]) by XXREAL_1:165,A1
.= ['a,c'] by INTEGRA5:def 3,XXREAL_0:2,A1;
 for x being object st x in ['a,c'] holds F . x in REAL
  by XREAL_0:def 1;
 hence thesis by FUNCT_2:3,Dg;
end;
