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;

theorem Th58:
  for pf being FinSequence of D holds
  k in dom pf implies <*pf*>*(1,k) = pf.k
  proof
    let pf be FinSequence of D;
    assume
A1: k in dom pf;
A2: Indices <*pf*> = [:Seg 1,Seg len pf:] by MATRIX_0:23
                  .= [:{1}, dom pf:] by FINSEQ_1:2,FINSEQ_1:def 3;
    1 in {1} by TARSKI:def 1;
    then [1,k] in Indices <*pf*> by A1,A2,ZFMISC_1:87;
    then ex q be FinSequence of D st q = <*pf*>.1 &
      <*pf*>*(1,k) = q.k by MATRIX_0:def 5;
    hence thesis;
  end;
