
theorem MM3:
for g be Function of REAL,REAL holds
{f where f is Function of REAL,REAL :
for x be Real holds f.x= min(1,max(0, g.x))}
 c= Membership_Funcs (REAL)
proof
 let g be Function of REAL,REAL;
 let f0 be object;
 assume f0 in {f where f is Function of REAL,REAL :
 for x be Real holds f.x= min(1,max(0, g.x))};
 then consider f be Function of REAL,REAL such that
 A1: f0=f and
 A2: for x be Real holds f.x= min(1,max(0, g.x));
 rng f c= [.0,1.]
 proof
  let y be object;
  assume y in rng f;then
  consider x be object such that
  B2: x in REAL and B3: y = f.x by FUNCT_2:11;
  reconsider x as Real by B2;
  B4:y = min(1,max(0, g.x)) by A2,B3;
  0<=max(0, g.x) by XXREAL_0:25; then
  0<=min(1,max(0, g.x)) & min(1,max(0, g.x))<=1 by XXREAL_0:20,XXREAL_0:17;
  hence thesis by B4;
 end;
 then f is [.0,1.] -valued;
 hence thesis by Def1,A1;
end;
