reserve D for non empty set,
  i,j,k,l for Nat,
  n for Nat,
  x for set,
  a,b,c,r,r1,r2 for Real,
  p,q for FinSequence of REAL,
  MR,MR1 for Matrix of REAL;

theorem Th20:
  for j st j in dom p holds Col(LineVec2Mx p,j) = <*p.j*>
proof
  set M = LineVec2Mx p;
  let j such that
A1: j in dom p;
A2: dom <*p.j*> = Seg 1 by FINSEQ_1:def 8;
A3: len Col(M,j) = len M by MATRIX_0:def 8;
  then len Col(M,j) = 1 by MATRIXR1:def 10;
  then
A4: dom Col(M,j) = dom <*p.j*> by A2,FINSEQ_1:def 3;
  now
    let k be Nat such that
A5: k in dom Col(M,j);
A6: k <= 1 by A2,A4,A5,FINSEQ_1:1;
    k >= 1 by A2,A4,A5,FINSEQ_1:1;
    then
A7: k = 1 by A6,XXREAL_0:1;
    k in dom M by A3,A5,FINSEQ_3:29;
    hence (Col(M,j)).k = M*(k,j) by MATRIX_0:def 8
      .= p.j by A1,A7,MATRIXR1:def 10
      .= (<*p.j*>).k by A7;
  end;
  hence thesis by A4,FINSEQ_1:13;
end;
