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 Th18:
  for T being RealLinearSpace
  for x,y being Element of T
  for p,q being Tuple of 1,REAL 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 Tuple of 1,REAL;
    assume that
A1: T = TOP-REAL 1 and
A2: p = x and
A3: q = y;
    set p9 = p, q9 = q;
    reconsider y9 = -y as Element of T;
    reconsider qm9 = -q as Tuple of 1,REAL by Th15;
A4: p9 in Funcs(Seg 1,REAL) &
      qm9 is Element of Funcs(Seg 1,REAL) by SRINGS_5:11;
A5:    (the addF of the RLSStruct of TOP-REAL 1).(p9,qm9)
      = (the addF of RealVectSpace Seg 1).(p9,qm9) by EUCLID:def 8
     .= p9 + qm9 by A4,FUNCSDOM:def 1;
    x - y = (the addF of (TOP-REAL 1)).(x,-y) by A1,ALGSTR_0:def 1
         .= p - q by A5,A1,A2,A3,Th16;
    hence thesis;
  end;
