reserve i,j,k for Nat;
reserve K for non empty addLoopStr,
  a for Element of K,
  p for FinSequence of the carrier of K,
  R for Element of i-tuples_on the carrier of K;
reserve K for left_zeroed right_zeroed add-associative right_complementable
  non empty addLoopStr,
  R,R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for non empty addLoopStr,
  a1,a2 for Element of K,
  p1,p2 for FinSequence of the carrier of K,
  R1,R2 for Element of i-tuples_on the carrier of K;
reserve K for Abelian right_zeroed add-associative right_complementable non
  empty addLoopStr,
  R,R1,R2,R3 for Element of i-tuples_on the carrier of K;
reserve K for non empty multMagma,
  a,a9,a1,a2 for Element of K,
  p for FinSequence of the carrier of K,
  R for Element of i-tuples_on the carrier of K;

theorem Th50:
  i in dom (a*p) & a9 = p.i implies (a*p).i = a*a9
proof
  assume
A1: i in dom (a*p) & a9 = p.i; then
A2: a9 in dom((the multF of K)[;](a,id the carrier of K)) by FUNCT_1:11;
  thus (a*p).i = ((the multF of K)[;](a,id the carrier of K)).a9 by A1,
FUNCT_1:12
    .=(the multF of K).(a,(id the carrier of K).a9) by A2,FUNCOP_1:32
    .= a*a9;
end;
