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 a being Real for p,q being Element of TOP-REAL 1 st
  T = TOP-REAL 1 & p = x & q = y holds x = a * y implies p = a * q
  proof
    let T be RealLinearSpace;
    let x,y be Element of T;
    let a be Real;
    let p,q be Element of TOP-REAL 1;
    assume that
A1: T = TOP-REAL 1 and
A2: p = x and
A3: q = y and
A4: x = a * y;
    p is Element of REAL 1 & q is Element of REAL 1 by EUCLID:22;
    then reconsider p9 = p,q9 = q as Tuple of 1,REAL;
    p9 = a * q9 by A1,A2,A3,A4,Th23;
    hence thesis;
  end;
