reserve i,j for Element of NAT,
  x,y,z for FinSequence of COMPLEX,
  c for Element of COMPLEX,
  R,R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th34:
  for c being Complex holds c*(0c (len x)) = 0c (len x)
proof
  let c be Complex;
  reconsider cc = c as Element of COMPLEX by XCMPLX_0:def 2;
A1: rng (0c (len x)) c= COMPLEX by FINSEQ_1:def 4;
  c*(0c (len x)) = multcomplex[;](cc,id COMPLEX)*(0c (len x)) by Lm1
    .= multcomplex[;](cc,(id COMPLEX)*(0c (len x))) by FUNCOP_1:34
    .= multcomplex[;](cc,(0c (len x))) by A1,RELAT_1:53
    .= multcomplex[;](cc,(len x |-> 0c)) by SEQ_4:def 12
    .= (len x) |->(multcomplex.(cc,0c)) by FINSEQOP:18
    .= (len x) |->(cc*0c) by BINOP_2:def 5
    .= (0c (len x)) by SEQ_4:def 12;
  hence thesis;
end;
