reserve V for non empty RLSStruct;
reserve x,y,y1 for set;
reserve v for VECTOR of V;
reserve a,b for Real;
reserve V for non empty addLoopStr;
reserve F for FinSequence-like PartFunc of NAT,V;
reserve f,f9,g for sequence of V;
reserve v,u for Element of V;
reserve j,k,n for Nat;
reserve V for RealLinearSpace;
reserve v for VECTOR of V;
reserve F,G,H,I for FinSequence of V;
reserve V for add-associative right_zeroed right_complementable non empty
  addLoopStr;
reserve F for FinSequence of V;
reserve v,v1,v2,u,w for Element of V;
reserve j,k for Nat;

theorem Th46:
  for V being add-associative right_zeroed right_complementable
  non empty addLoopStr, v,u,w being Element of V holds
  Sum<* v,u,w *> = v + u + w
proof
  let V be add-associative right_zeroed right_complementable non empty
  addLoopStr, v,u,w be Element of V;
  <* v,u,w *> = <* v,u *> ^ <* w *> by FINSEQ_1:43;
  hence Sum<* v,u,w *> = Sum<* v,u *> + Sum<* w *> by Th41
    .= Sum<* v,u *> + w by Lm6
    .= v + u + w by Th45;
end;
