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 Th30:
  for M being Matrix of D for p being FinSequence of D* st len p =
len M & p.1 = M.1 & (for k st k >= 1 & k < len M holds p.(k+1) = (p.k) ^ M.(k+1
)) holds for i,j st i in dom p & j in dom p & i <= j holds dom (p.i) c= dom (p.
  j)
proof
  let M be Matrix of D;
  let p be FinSequence of D* such that
A1: len p = len M and
A2: p.1 = M.1 and
A3: for k st k >= 1 & k < len M holds p.(k+1) = (p.k) ^ M.(k+1);
  let i,j such that
A4: i in dom p and
A5: j in dom p and
A6: i <= j;
A7: len (p.j) = j * width M by A1,A2,A3,A5,Th29;
  len (p.i) = i * width M by A1,A2,A3,A4,Th29;
  then len (p.i) <= len (p.j) by A6,A7,NAT_1:4;
  then Seg len (p.i) c= Seg len (p.j) by FINSEQ_1:5;
  then dom (p.i) c= Seg len (p.j) by FINSEQ_1:def 3;
  hence thesis by FINSEQ_1:def 3;
end;
