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 Th66:
  for p being FinSequence of 1-tuples_on REAL st len p = 3 holds
  M2F p is Point of TOP-REAL 3
  proof
    let p being FinSequence of 1-tuples_on REAL;
    assume
A1: len p = 3;
A2: dom p = Seg 3 by A1,FINSEQ_1:def 3; then
A3: p.1 in rng p by FINSEQ_1:1,FUNCT_1:3;
A4: rng p c= 1-tuples_on REAL by FINSEQ_1:def 4;
    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 A3,A4;
    then consider d be Element of REAL such that
A5: p.1 = <*d*>;
A6: (p.1).1 = d by A5;
A7: p.2 in rng p by A2,FINSEQ_1:1,FUNCT_1:3;
    1-tuples_on REAL = the set of all <*d*> where d is Element of REAL
      by FINSEQ_2:96;
    then p.2 in the set of all <*d*> where d is Element of REAL by A7,A4;
    then consider d be Element of REAL such that
A9: p.2 = <*d*>;
A10: (p.2).1 = d by A9;
A11: p.3 in rng p by A2,FINSEQ_1:1,FUNCT_1:3;
    rng p c= 1-tuples_on REAL by FINSEQ_1:def 4; then
A12: p.3 in 1-tuples_on REAL by A11;
    1-tuples_on REAL = the set of all <*d*> where d is Element of REAL
      by FINSEQ_2:96;
    then consider d be Element of REAL such that
A13: p.3 = <*d*> by A12;
A14: (p.3).1 = d by A13;
    M2F p = <* (p.1).1, (p.2).1, (p.3).1 *> by A1,DEF2;
    then M2F p in 3-tuples_on REAL by A6,A10,A14,FINSEQ_2:104;
    then M2F p in REAL 3 by EUCLID:def 1;
    hence thesis by EUCLID:22;
  end;
