reserve X for non empty set;
reserve Y for RealLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Real;
reserve u,v,w for VECTOR of RLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem Th22:
  for X, Y be RealNormSpace holds BoundedLinearOperators(X,Y) is
  linearly-closed
proof
  let X, Y be RealNormSpace;
  set W = BoundedLinearOperators(X,Y);
A1: for v,u be VECTOR of R_VectorSpace_of_LinearOperators(X, Y) st v in W &
  u in W holds v + u in W
  proof
    let v,u be VECTOR of R_VectorSpace_of_LinearOperators(X, Y) such that
A2: v in W and
A3: u in W;
    reconsider f=v+u as LinearOperator of X,Y by Def6;
    f is Lipschitzian
    proof
      reconsider v1=v as Lipschitzian LinearOperator of X,Y by A2,Def9;
      consider K2 be Real such that
A4:   0 <= K2 and
A5:   for x be VECTOR of X holds ||. v1.x .|| <= K2*||. x .|| by Def8;
      reconsider u1=u as Lipschitzian LinearOperator of X,Y by A3,Def9;
      consider K1 be Real such that
A6:   0 <= K1 and
A7:   for x be VECTOR of X holds ||. u1.x .|| <= K1*||. x .|| by Def8;
      take K3=K1+K2;
      now
        let x be VECTOR of X;
A8:     ||. u1.x+v1.x .|| <= ||. u1.x .||+ ||. v1.x .|| by NORMSP_1:def 1;
A9:     ||. v1.x .|| <= K2*||. x .|| by A5;
        ||. u1.x .|| <= K1*||. x .|| by A7;
        then
A10:    ||. u1.x .|| + ||. v1.x .|| <= K1*||. x .|| +K2*||. x .|| by A9,
XREAL_1:7;
        ||. f.x .|| =||. u1.x+v1.x .|| by Th16;
        hence ||. f.x .|| <= K3*||. x .|| by A8,A10,XXREAL_0:2;
      end;
      hence thesis by A6,A4;
    end;
    hence thesis by Def9;
  end;
  for a be Real
  for v be VECTOR of R_VectorSpace_of_LinearOperators(X,Y)
  st v in W holds a * v in W
  proof
    let a be Real;
    let v be VECTOR of R_VectorSpace_of_LinearOperators(X,Y) such that
A11: v in W;
    reconsider f=a*v as LinearOperator of X,Y by Def6;
    f is Lipschitzian
    proof
      reconsider v1=v as Lipschitzian LinearOperator of X,Y by A11,Def9;
      consider K be Real such that
A12:  0 <= K and
A13:  for x be VECTOR of X holds ||. v1.x .|| <= K*||. x .|| by Def8;
      take |.a.|*K;
A14:  now
        let x be VECTOR of X;
        0 <=|.a.| by COMPLEX1:46;
        then
A15:    |.a.|* ||. v1.x .|| <= |.a.|* (K*||. x .||) by A13,XREAL_1:64;
        ||. a*v1.x .|| = |.a.|* ||. v1.x .|| by NORMSP_1:def 1;
        hence ||. f.x .|| <= |.a.|* K*||. x .|| by A15,Th17;
      end;
      0 <=|.a.| by COMPLEX1:46;
      hence thesis by A12,A14;
    end;
    hence thesis by Def9;
  end;
  hence thesis by A1,RLSUB_1:def 1;
end;
