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 Th14:
  for X, Y be RealLinearSpace holds LinearOperators(X,Y) is linearly-closed
proof
  let X, Y be RealLinearSpace;
  set W = LinearOperators(X,Y);
A1: for v,u be VECTOR of RealVectSpace(the carrier of X, Y) st v in
LinearOperators(X,Y) & u in LinearOperators(X,Y) holds v + u in LinearOperators
  (X,Y)
  proof
    let v,u be VECTOR of RealVectSpace(the carrier of X,Y) such that
A2: v in W and
A3: u in W;
    v+u is LinearOperator of X,Y
    proof
      reconsider f=v+u as Function of X,Y by FUNCT_2:66;
A4:   f is homogeneous
      proof
        reconsider v9 = v, u9 = u as Element of Funcs(the carrier of X,the
        carrier of Y);
        let x be VECTOR of X, s be Real;
A5:     u9 is LinearOperator of X,Y by A3,Def6;
A6:     v9 is LinearOperator of X,Y by A2,Def6;
        thus f.(s*x) =u9.(s*x)+v9.(s*x) by Th1
          .=(s*(u9.x))+v9.(s*x) by A5,Def5
          .=s*u9.x+s*v9.x by A6,Def5
          .=s*(u9.x+v9.x) by RLVECT_1:def 5
          .= s*f.x by Th1;
      end;
      f is additive
      proof
        reconsider v9 = v, u9 = u as Element of Funcs(the carrier of X,the
        carrier of Y);
        let x,y be VECTOR of X;
A7:     u9 is LinearOperator of X,Y by A3,Def6;
A8:     v9 is LinearOperator of X,Y by A2,Def6;
        thus f.(x+y) =u9.(x+y)+v9.(x+y) by Th1
          .=(u9.x+u9.y)+v9.(x+y) by A7,VECTSP_1:def 20
          .=u9.x+u9.y+(v9.x+v9.y) by A8,VECTSP_1:def 20
          .=u9.x+u9.y+v9.x+v9.y by RLVECT_1:def 3
          .=u9.x+v9.x+u9.y+v9.y by RLVECT_1:def 3
          .= f.x+ u9.y+v9.y by Th1
          .= f.x+ (u9.y+v9.y) by RLVECT_1:def 3
          .= f.x+f.y by Th1;
      end;
      hence thesis by A4;
    end;
    hence thesis by Def6;
  end;
  for a be Real
  for v be VECTOR of RealVectSpace(the carrier of X,Y) st v
  in W holds a * v in W
  proof
    let a be Real;
    let v be VECTOR of RealVectSpace(the carrier of X,Y) such that
A9: v in W;
    a*v is LinearOperator of X,Y
    proof
      reconsider f=a*v as Function of X,Y by FUNCT_2:66;
A10:  f is homogeneous
      proof
        reconsider v9 = v as Element of Funcs(the carrier of X,the carrier of
        Y);
        let x be VECTOR of X, s be Real;
A11:    v9 is LinearOperator of X,Y by A9,Def6;
        thus f.(s*x) =a*v9.(s*x) by Th2
          .=a*(s*(v9.x)) by A11,Def5
          .=a*s*v9.x by RLVECT_1:def 7
          .=s*(a*v9.x) by RLVECT_1:def 7
          .=s*f.x by Th2;
      end;
      f is additive
      proof
        reconsider v9 = v as Element of Funcs(the carrier of X,the carrier of
        Y);
        let x,y be VECTOR of X;
A12:    v9 is LinearOperator of X,Y by A9,Def6;
        thus f.(x+y) = a*(v9.(x+y)) by Th2
          .=a*(v9.x+v9.y) by A12,VECTSP_1:def 20
          .=a*v9.x+a*v9.y by RLVECT_1:def 5
          .=f.x+a*v9.y by Th2
          .=f.x+ f.y by Th2;
      end;
      hence thesis by A10;
    end;
    hence thesis by Def6;
  end;
  hence thesis by A1,RLSUB_1:def 1;
end;
