reserve x,y,y1,y2 for set,
  p for FinSequence,
  i,k,l,n for Nat,
  V for RealLinearSpace,
  u,v,v1,v2,v3,w for VECTOR of V,
  a,b for Real,
  F,G,H1,H2 for FinSequence of V,
  A,B for Subset of V,
  f for Function of the carrier of V, REAL;

theorem
  for V be Abelian add-associative right_zeroed right_complementable
non empty addLoopStr, v1,v2,v3 be Element of V holds v1 <> v2 & v2 <> v3 & v1
  <> v3 implies Sum{v1,v2,v3} = v1 + v2 + v3
proof
  let V be Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr, v1,v2,v3 be Element of V;
  assume v1 <> v2 & v2 <> v3 & v1 <> v3;
  then
A1: <* v1,v2,v3 *> is one-to-one by FINSEQ_3:95;
  rng<* v1,v2,v3 *> = {v1,v2,v3} & Sum<* v1,v2,v3 *> = v1 + v2 + v3 by
FINSEQ_2:128,RLVECT_1:46;
  hence thesis by A1,Def2;
end;
