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

theorem Th17:
  for X,Y be ComplexLinearSpace holds 0.
  C_VectorSpace_of_LinearOperators(X,Y) = (the carrier of X) -->0.Y
proof
  let X,Y be ComplexLinearSpace;
A1: 0.ComplexVectSpace(the carrier of X,Y) =((the carrier of X) -->0.Y) by
LOPBAN_1:def 3;
  C_VectorSpace_of_LinearOperators(X,Y) is Subspace of ComplexVectSpace(
  the carrier of X,Y) by Th13,CSSPACE:11;
  hence thesis by A1,CLVECT_1:30;
end;
