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 Th65:
  for V being Abelian add-associative right_zeroed
  right_complementable non empty addLoopStr, v,u,w being Element of V holds
  Sum <* u,v,w *> = Sum<* u,w *> + v
proof
  let V be Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr, v,u,w be Element of V;
  thus Sum<* u,v,w *> = u + v + w by Th46
    .= u + w + v by Def3
    .= Sum<* u,w *> + v by Th45;
end;
