
theorem
  for r be Real, f,g be Function of REAL,REAL holds
    r (#) (f +* g) = (r (#) f) +* (r (#) g)
proof
 let r be Real, f,g be Function of REAL,REAL;
 set F1 = r (#) (f +* g);
 set F2 = (r (#) f) +* (r (#) g);
 D1:dom F1 = REAL & dom F2 = REAL by FUNCT_2:def 1;
 for x being object st x in dom F1 holds F1 . x = F2 . x
 proof
  let x be object;
  assume A1: x in dom F1; then
  reconsider x as Real;
  A2: x in REAL by A1;
  per cases;
  suppose B1: x in dom g;
   B2: x in dom (r (#) g) by FUNCT_2:def 1,A2;
   (r (#) (f +* g)).x = r * (f +* g).x by VALUED_1:6
    .= r * ( g).x by FUNCT_4:13,B1
    .= (r (#) g).x by VALUED_1:6
   .= ((r (#) f) +* (r (#) g)).x by FUNCT_4:13,B2;
   hence thesis;
  end;
  suppose C1: not x in dom g;
   C2: dom g = REAL by FUNCT_2:def 1
     .= dom (r (#) g) by FUNCT_2:def 1;
   (r (#) (f +* g)).x = r * (f +* g).x by VALUED_1:6
    .= r * f.x by FUNCT_4:11,C1
    .= (r (#) f).x by VALUED_1:6
   .= ((r (#) f) +* (r (#) g)).x by FUNCT_4:11,C2,C1;
   hence thesis;
  end;
 end;
 hence thesis by FUNCT_1:2,D1;
end;
