
theorem Th3:
  for z being Complex, v being VECTOR of Linear_Space_of_ComplexSequences holds
  z * v = z(#)seq_id(v)
proof
  let z be Complex;
  let v be VECTOR of Linear_Space_of_ComplexSequences;
  reconsider r1 = z as Element of COMPLEX by XCMPLX_0:def 2;
  reconsider v1 = v as Element of Funcs(NAT,COMPLEX);
  reconsider h = (ComplexFuncExtMult NAT).(r1,v1) as Complex_Sequence
  by FUNCT_2:66;
  h = z(#)seq_id(v)
  proof
    let n be Element of NAT;
    thus h.n = z*(v1.n) by CFUNCDOM:4
    .= (z(#)seq_id(v)).n by VALUED_1:6;
  end;
  hence thesis;
end;
