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 Th25:
  for c being Complex, x,y being FinSequence of COMPLEX st
  len x=len y holds c*(x+y) = c*x + c*y
proof
  let c be Complex, x,y be FinSequence of COMPLEX;
  assume
A1: len x=len y;
  set cM = c multcomplex;
  reconsider y9=y as Element of (len y)-tuples_on COMPLEX by FINSEQ_2:92;
  reconsider x9=x as Element of (len x)-tuples_on COMPLEX by FINSEQ_2:92;
A2: c is Element of COMPLEX by XCMPLX_0:def 2;
  c*(x+y) = cM*(x + y) by SEQ_4:def 9
    .= cM*(addcomplex.:(x,y)) by SEQ_4:def 6
    .= addcomplex.:(cM*x9,cM*y9) by A1,A2,FINSEQOP:51,SEQ_4:56
    .= cM*x + cM*y by SEQ_4:def 6
    .= c*x + cM*y by SEQ_4:def 9
    .= c*x + c*y by SEQ_4:def 9;
  hence thesis;
end;
