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 Th59:
  k in dom pf implies Col(<*pf*>,k) = <*pf.k*>
  proof
    assume
A1: k in dom pf;
A2: len Col(<*pf*>,k) = len <*pf*> & for j be Nat st j in dom <*pf*> holds
      Col(<*pf*>,k).j = <*pf*>*(j,k) by MATRIX_0:def 8; then
A3: len Col(<*pf*>,k) = 1 by FINSEQ_1:39; then
A4: dom Col(<*pf*>,k) = {1} by FINSEQ_1:def 3,FINSEQ_1:2;
    (Col(<*pf*>,k)).1 = <*pf*>*(1,k) by FINSEQ_5:6,A2;
    then rng Col(<*pf*>,k) = {<*pf*>*(1,k)} by A4,FUNCT_1:4;
    then rng Col(<*pf*>,k) = {pf.k} by A1,Th58;
    hence thesis by A3,FINSEQ_1:39;
  end;
