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 Th63:
  len pf = 3 implies <*pf*>@ = <* <* pf.1 *>, <* pf.2 *>, <* pf.3 *> *>
  proof
    assume
A1: len pf = 3;
A2: len <*pf*> = 1 by FINSEQ_1:39;
    rng <*pf*> = {pf} by FINSEQ_1:39;
    then pf in rng <*pf*> by TARSKI:def 1; then
A3: width <*pf*> = 3 by A1,A2,MATRIX_0:def 3; then
A4: width (<*pf*>@) = len <*pf*> by MATRIX_0:29
                   .= 1 by FINSEQ_1:39;
    now
      thus len (<*pf*>@) = 3 by MATRIX_0:def 6,A3; then
A5:   <*pf*>@ is Matrix of 3,1,D by A4,MATRIX_0:20;
      1 in Seg 3 by FINSEQ_1:1;
      hence <*pf*>@.1 = Line (<*pf*>@,1) by A5,MATRIX_0:52
                     .= <* pf.1 *> by A1,Th62;
      2 in Seg 3 by FINSEQ_1:1;
      hence <*pf*>@.2 = Line (<*pf*>@,2) by A5,MATRIX_0:52
                    .= <* pf.2 *> by A1,Th62;
      3 in Seg 3 by FINSEQ_1:1;
      hence <*pf*>@.3 = Line (<*pf*>@,3) by A5,MATRIX_0:52
                    .= <* pf.3 *> by A1,Th62;
    end;
    hence <*pf*>@ = <* <* pf.1 *>, <* pf.2 *>, <* pf.3 *> *> by FINSEQ_1:45;
end;
