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 Th14:
  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;
A4: p in Funcs(Seg 1,REAL) & q is Element of Funcs(Seg 1,REAL) by SRINGS_5:11;
    (the addF of the RLSStruct of TOP-REAL 1).(p,q)
      = (the addF of RealVectSpace Seg 1).(p,q) by EUCLID:def 8
     .= p + q by A4,FUNCSDOM:def 1;
    hence thesis by A1,A2,A3,ALGSTR_0:def 1;
  end;
