reserve X for non empty set;
reserve x for Element of X;
reserve d1,d2 for Element of X;
reserve A for BinOp of X;
reserve M for Function of [:X,X:],X;
reserve V for Ring;
reserve V1 for Subset of V;
reserve V for Algebra;
reserve V1 for Subset of V;
reserve MR for Function of [:REAL,X:],X;
reserve a for Real;

theorem Th9:
  BoundedFunctions X is additively-linearly-closed multiplicatively-closed
proof
  set W = BoundedFunctions X;
  set V = RAlgebra X;
A1: for v,u be Element of V st v in W & u in W holds v * u in W
  proof
    let v,u be Element of V such that
A2: v in W and
A3: u in W;
    consider u1 be Function of X,REAL such that
A4: u1=u and
A5: u1|X is bounded by A3;
    reconsider h = v*u as Element of Funcs(X,REAL);
    consider v1 be Function of X,REAL such that
A6: v1=v and
A7: v1|X is bounded by A2;
    dom(v1(#)u1) = X /\ X by FUNCT_2:def 1;
    then
A8: (v1(#)u1)|X is bounded by A7,A5,RFUNCT_1:84;
    dom v1 /\ dom u1 = X /\ dom u1 by FUNCT_2:def 1;
    then
A9: (ex f being Function st h = f & dom f = X & rng f c= REAL)& dom v1 /\
    dom u1 = X /\ X by FUNCT_2:def 1,def 2;
    for x be object st x in dom h holds h.x = v1.x *u1.x by A6,A4,FUNCSDOM:2;
    then v*u=v1(#)u1 by A9,VALUED_1:def 4;
    hence thesis by A8;
  end;
  reconsider g = RealFuncUnit X as Function of X,REAL;
  for v,u be Element of V st v in W & u in W holds v + u in W
  proof
    let v,u be Element of V such that
A10: v in W and
A11: u in W;
    consider u1 be Function of X,REAL such that
A12: u1=u and
A13: u1|X is bounded by A11;
    reconsider h = v+u as Element of Funcs(X,REAL);
    consider v1 be Function of X,REAL such that
A14: v1=v and
A15: v1|X is bounded by A10;
    dom(v1+u1) = X /\ X by FUNCT_2:def 1;
    then
A16: (v1+u1)|X is bounded by A15,A13,RFUNCT_1:83;
    dom v1 /\ dom u1 = X /\ dom u1 by FUNCT_2:def 1;
    then
A17: (ex f being Function st h = f & dom f = X & rng f c= REAL)& dom v1 /\
    dom u1 = X /\ X by FUNCT_2:def 1,def 2;
    for x be object st x in dom h holds h.x = v1.x + u1.x
by A14,A12,FUNCSDOM:1;
    then v+u=v1+u1 by A17,VALUED_1:def 1;
    hence thesis by A16;
  end;
  then
A18: W is add-closed by IDEAL_1:def 1;
A19: RAlgebra X is RealLinearSpace by Th7;
  for v be Element of V st v in W holds -v in W
  proof
    let v be Element of V;
    assume v in W;
    then consider v1 be Function of X,REAL such that
A20: v1=v and
A21: v1|X is bounded;
A22: (-v1)|X is bounded by A21,RFUNCT_1:82;
    reconsider h = -v, v2 = v as Element of Funcs(X,REAL);
A23: h=(-1)*v by A19,RLVECT_1:16;
A24: now
      let x be object;
      assume x in dom h;
      then reconsider y=x as Element of X;
      h.x = (-1)*(v2.y) by A23,FUNCSDOM:4;
      hence h.x = -v1.x by A20;
    end;
    (ex f being Function st h = f & dom f = X & rng f c= REAL)& dom v1 =X
    by FUNCT_2:def 1,def 2;
    then -v=-v1 by A24,VALUED_1:9;
    hence thesis by A22;
  end;
  then
A25: W is having-inverse;
  for a be Real, u be Element of V st u in W holds a * u in W
  proof
    let a be Real, u be Element of V;
    assume u in W;
    then consider u1 be Function of X,REAL such that
A26: u1=u and
A27: u1|X is bounded;
A28: (a(#)u1)|X is bounded by A27,RFUNCT_1:80;
    reconsider h = a*u as Element of Funcs(X,REAL);
A29: (ex f being Function st h = f & dom f = X & rng f c= REAL)& dom u1 =
    X by FUNCT_2:def 1,def 2;
    for x be object st x in dom h holds h.x = a*(u1.x) by A26,FUNCSDOM:4;
    then a*u=a(#)u1 by A29,VALUED_1:def 5;
    hence thesis by A28;
  end;
  hence BoundedFunctions X is additively-linearly-closed by A18,A25;
  g|X is bounded;
  then 1.V in W;
  hence thesis by A1;
end;
