
theorem LMThGM25:
  for i, j being Nat
  for K being Field
  for aj, bj being Element of K
  for A,B being Element of i-VectSp_over K
  st j in Seg i & aj = A.j & bj = B.j holds
  (A+B).j = aj + bj
  proof
    let i, j be Nat;
    let K be Field;
    let aj, bj be Element of K;
    let A, B be Element of i-VectSp_over K;
    assume AS: j in Seg i & aj = A.j & bj = B.j;
    P1: the addLoopStr of i-VectSp_over K = i-Group_over K by PRVECT_1:def 5;
    P2:i-Group_over K = addLoopStr(# i-tuples_on the carrier of K,
    product(the addF of K,i), (i |-> 0.K) qua Element
    of i-tuples_on the carrier of K#) by PRVECT_1:def 3;
    P0: the carrier of i-VectSp_over K = i-tuples_on the carrier of K
    by MATRIX13:102;
    reconsider A0 = A, B0 = B as Element of i-tuples_on the carrier of K
    by MATRIX13:102;
    P3: A+B = (the addF of K).:(A0,B0) by P1,P2,PRVECT_1:def 1;
    A+B in i -tuples_on  the carrier of K by P0;
    then consider s be Element of (the carrier of K)* such that
    P4: A+B = s & len s = i;
    dom ((the addF of K).:(A0,B0)) = Seg i by P3,P4,FINSEQ_1:def 3;
    hence (A+B).j = aj+bj by P3,AS,FUNCOP_1:22;
  end;
