reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);
reserve o,p,q,r,s,t for Point of TOP-REAL 3,
        M for Matrix of 3,F_Real;
reserve pf for FinSequence of D;
reserve PQR for Matrix of 3,F_Real;
reserve R for Ring;

theorem Th81:
  for u being Element of TOP-REAL 3
  for uf being FinSequence of F_Real
  st u = uf & <*uf*>@ = <* <* 0 *>, <* 0 *>, <* 0 *> *>
  holds u = 0.TOP-REAL 3
  proof
    let u be Element of TOP-REAL 3;
    let uf be FinSequence of F_Real;
    assume that
A1: u = uf and
A2: <*uf*>@ = <* <* 0 *>, <* 0 *>, <* 0 *> *>;
    u in TOP-REAL 3;
    then u in REAL 3 by EUCLID:22; then
A3: u in 3-tuples_on REAL by EUCLID:def 1; then
A4: len uf = 3 by A1,FINSEQ_2:133;
    <*uf*>@ = <* <* uf.1 *>, <* uf.2 *> , <* uf.3 *> *>
      by A3,A1,FINSEQ_2:133,Th63;
     then <* uf.1 *> = <* 0 *> & <* uf.2 *> = <* 0 *> &
       <* uf.3 *> = <* 0 *> by A2,FINSEQ_1:78;
     then uf.1 = 0 & uf.2 = 0 & uf.3 = 0 by FINSEQ_1:76;
     hence thesis by A1,A4,FINSEQ_1:45,EUCLID_5:4;
   end;
