
theorem LMThGM23:
  for n being Nat, r being FinSequence of F_Rat st len r = n
  holds
  ex K being Integer, Kr being FinSequence of INT.Ring
  st K <> 0 & len Kr = n
  & for i being Nat st i in dom Kr
  holds Kr.i = K * r/.i
  proof
    let n be Nat, r be FinSequence of F_Rat;
    assume len r = n;
    then consider K be Integer such that
    P1: K <> 0 &
    for i being Nat st i in Seg n holds K * r/.i in INT by LMThGM231;
    defpred Q[Nat,object] means $2 = K * (r/.$1);
    Z510: for i being Nat st i in Seg n
    ex x being Element of the carrier of INT.Ring st Q[i,x]
    proof
      let i be Nat;
      assume i in Seg n;
      then reconsider x = K*r/.i as Element of INT.Ring by P1;
      take x;
      thus thesis;
    end;
    consider Kr be FinSequence of the carrier of INT.Ring such that
    Z511: dom Kr = Seg n &
    for k being Nat st k in Seg n holds Q[k,Kr.k] from FINSEQ_1:sch 5(Z510) ;
    take K, Kr;
    thus K <> 0 by P1;
    n is Element of NAT by ORDINAL1:def 12;
    hence len Kr = n by Z511,FINSEQ_1:def 3;
    thus for i being Nat st i in dom Kr holds Kr.i = K * r/.i by Z511;
  end;
