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;

theorem Th68:
  for p being FinSequence of 1-tuples_on REAL st len p = 3 holds
  <* <*(p.1).1*>, <*(p.2).1*> , <*(p.3).1*> *> = p
  proof
    let p be FinSequence of 1-tuples_on REAL;
    assume
A1: len p = 3;
    then dom p = Seg 3 by FINSEQ_1:def 3; then
A2: p.1 in rng p & p.2 in rng p & p.3 in rng p by FINSEQ_1:1,FUNCT_1:3;
A3: rng p c= 1-tuples_on REAL by FINSEQ_1:def 4;
A4: 1-tuples_on REAL = the set of all <*d*> where d is Element of REAL
      by FINSEQ_2:96;
    then p.1 in the set of all <*d*> where d is Element of REAL by A2,A3;
    then consider d1 be Element of REAL such that
A5: p.1 = <*d1*>;
    p.2 in the set of all <*d*> where d is Element of REAL by A2,A3,A4;
    then consider d2 be Element of REAL such that
A7: p.2 = <*d2*>;
    p.3 in the set of all <*d*> where d is Element of REAL by A2,A3,A4;
    then consider d3 be Element of REAL such that
A9: p.3 = <*d3*>;
    thus thesis by A5,A7,A9,A1,FINSEQ_1:45;
  end;
