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 Th23:
  for T being RealLinearSpace for x,y being Element of T
  for a being Real for p,q being Tuple of 1,REAL st
  T = TOP-REAL 1 & p = x & q = y & x = a * y holds p = a * q
  proof
    let T be RealLinearSpace;
    let x,y be Element of T;
    let a be Real;
    let p,q be Tuple of 1,REAL;
    assume that
A1: T = TOP-REAL 1 and
A2: p = x and
A3: q = y and
A4: x = a * y;
    set p9 = q;
A5: p9 in Funcs(Seg 1,REAL) by SRINGS_5:11;
    (the Mult of the RLSStruct of TOP-REAL 1).(a,p9)
      = (the Mult of RealVectSpace Seg 1).(a,p9) by EUCLID:def 8
     .= multreal[;](a,p9) by A5,FUNCSDOM:def 3
     .= multreal.:(dom p9 --> a,p9) by FUNCOP_1:31
     .= a * p9 by Th22;
    hence thesis by A1,A2,A3,A4;
  end;
