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;
reserve x,y for set,
  k,n for Element of NAT;

theorem
  for V being Abelian add-associative right_zeroed right_complementable
  non empty addLoopStr, v,w being Element of V holds
  Sum<* v,- w *> = v - w & Sum<* - w,v *> = v - w
proof
  let V be Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr, v,w be Element of V;
  thus Sum<* v,- w *> = v - w by Th45;
  hence thesis by Th54;
end;
