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 Th19:
  for X,Y be RealLinearSpace holds (the carrier of X) --> 0.Y is
  LinearOperator of X,Y
proof
  let X,Y be RealLinearSpace;
  set f=(the carrier of X) --> 0.Y;
  reconsider f as Function of X,Y;
A1: f is homogeneous
  proof
    let x be VECTOR of X, r be Real;
    thus f.(r*x) =0.Y by FUNCOP_1:7
      .=r*0.Y
      .= r*f.x by FUNCOP_1:7;
  end;
  f is additive
  proof
    let x,y be VECTOR of X;
    thus f.(x+y) = 0.Y by FUNCOP_1:7
      .=0.Y+0.Y
      .= f.x+0.Y by FUNCOP_1:7
      .= f.x+f.y by FUNCOP_1:7;
  end;
  hence thesis by A1;
end;
