reserve m,k,j,j1,i,i1,i2,n for Nat,
  r,s for Real,
  C for compact non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p for Point of TOP-REAL 2;

theorem Th13:
  for D being non empty set, M being (Matrix of D), i st i in dom
  M holds rng Line(M,i) c= Values M
proof
  let D be non empty set;
  let M be (Matrix of D), k;
  assume k in dom M;
  then
A1: 1 <= k & k <= len M by FINSEQ_3:25;
  let e be object;
  assume e in rng Line(M,k);
  then consider u being object such that
A2: u in dom Line(M,k) and
A3: e = Line(M,k).u by FUNCT_1:def 3;
  reconsider u as Nat by A2;
A4: 1 <= u by A2,FINSEQ_3:25;
A5: len Line(M,k) = width M by MATRIX_0:def 7;
  then u <= width M by A2,FINSEQ_3:25;
  then
A6: [k,u] in Indices M by A1,A4,MATRIX_0:30;
A7: Values M = { M*(i,j) where i,j is Nat: [i,j] in Indices M }
    by MATRIX_0:39;
  dom Line(M,k) = Seg width M by A5,FINSEQ_1:def 3;
  then Line(M,k).u = M*(k,u) by A2,MATRIX_0:def 7;
  hence thesis by A7,A3,A6;
end;
