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 x,y being Element of TOP-REAL 1 for a,b,r being Real st
  x = <* a *> & y = <* b *> holds x = r * y iff a = r * b
  proof
    let x,y be Element of TOP-REAL 1;
    let a,b,r be Real;
    assume that
A1: x = <* a *> and
A2: y = <* b *>;
    reconsider rb = r * b as Real;
    hereby
      assume x = r * y;
      then x = <* rb *> by A2,Th27;
      hence a = r * b by A1,FINSEQ_1:76;
    end;
    assume a = r * b;
    hence x = r * y by A2,A1,Th27;
  end;
