reserve a,b,r for non unit non zero Real;
reserve X for non empty set,
        x for Tuple of 4,X;
reserve V             for RealLinearSpace,
        A,B,C,P,Q,R,S for Element of V;

theorem
  for T being RealLinearSpace for x,y being Element of T
  for p,q being Element of TOP-REAL 1 st T = TOP-REAL 1 & p = x & q = y
  holds x - y = p - q
  proof
    let T be RealLinearSpace;
    let x,y be Element of T;
    let p,q be Element of TOP-REAL 1;
    assume that
A1: T = TOP-REAL 1 and
A2: p = x and
A3: q = y;
    reconsider y9 = -y as Element of T;
    reconsider q as Element of REAL 1 by EUCLID:22;
    reconsider q9 = -q as Element of TOP-REAL 1 by EUCLID:22;
    - q = - y by A1,A3,Th17;
    hence thesis by A1,A2,Th19;
  end;
