reserve A for non empty closed_interval Subset of REAL;

theorem Th25:
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 = F | [. a,c .]
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.];
 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;
 hence thesis;
end;
