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 Th24:
  for p,q being Tuple of 1,REAL for r being Real st
  p = r * q & p <> <* 0 *> holds ex a,b being Real st
  p = <* a *> & q =<* b *> & r = a / b
  proof
    let p,q be Tuple of 1,REAL;
    let r be Real;
    assume that
A1: p = r * q and
A2: p <> <* 0 *>;
    consider r1 be Element of REAL such that
A3: p = <* r1 *> by FINSEQ_2:97;
    consider r2 be Element of REAL such that
A4: q = <* r2 *> by FINSEQ_2:97;
    reconsider r1,r2 as Real;
    take r1,r2;
A5: <* r1 *> = <* r * r2 *> by A1,A3,A4,RVSUM_1:47;
    then
A6: r1 = r * r2 by FINSEQ_1:76;
    per cases;
    suppose
      r2 = 0;
      hence thesis by A5,A2,A3;
    end;
    suppose
A7:   r2 <> 0;
      r = r * 1
       .= r * (r2 / r2) by A7,XCMPLX_1:60
       .= r1 / r2 by A6;
      hence thesis by A3,A4;
    end;
  end;
