
theorem LMThGM24:
  for i, j being Nat
  for K being Field
  for a, aj being Element of K
  for R being Element of i-VectSp_over K
  st j in Seg i & aj = R.j holds
  (a * R).j = a * aj
  proof
    let i, j be Nat;
    let K be Field;
    let a, aj be Element of K;
    let R be Element of i-VectSp_over K;
    assume AS: j in Seg i & aj = R.j;
    P0: the carrier of i-VectSp_over K = i -tuples_on  the carrier of K
    by MATRIX13:102;
    reconsider R0 = R as Element of i -tuples_on the carrier of K
    by MATRIX13:102;
    P3: a * R = (i -Mult_over K).(a,R) by PRVECT_1:def 5
    .= (the multF of K)[;](a,R0) by PRVECT_1:def 4;
    a*R in i -tuples_on  the carrier of K by P0;
    then consider s be Element of (the carrier of K)* such that
    P4: a*R = s & len s = i;
    dom ((the multF of K)[;](a,R0)) = Seg i by P3,P4,FINSEQ_1:def 3;
    hence (a * R).j = a * aj by P3,AS,FUNCOP_1:32;
  end;
