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 Th18:
  for X, Y be RealLinearSpace holds 0.
  R_VectorSpace_of_LinearOperators(X,Y) = (the carrier of X) -->0.Y
proof
  let X, Y be RealLinearSpace;
A1: 0.RealVectSpace(the carrier of X,Y) =((the carrier of X) -->0.Y);
  R_VectorSpace_of_LinearOperators(X,Y) is Subspace of RealVectSpace(the
  carrier of X,Y) by Th14,RSSPACE:11;
  hence thesis by A1,RLSUB_1:11;
end;
