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 Th32:
  for M being Matrix of D for p being FinSequence of D* st len p =
len M & (for k st k >= 1 & k < len M holds p.(k+1) = (p.k) ^ M.(k+1)) holds for
j st j >= 1 & j < len p holds for l st l in dom(p.j) holds (p.j).l = (p.(j+1)).
  l
proof
  let M be Matrix of D;
  let p be FinSequence of D* such that
A1: len p = len M and
A2: for k st k >= 1 & k < len M holds p.(k+1) = (p.k) ^ M.(k+1);
  let j such that
A3: j >= 1 and
A4: j < len p;
A5: p.(j+1) = (p.j) ^ M.(j+1) by A1,A2,A3,A4;
  let l;
  assume l in dom(p.j);
  hence thesis by A5,FINSEQ_1:def 7;
end;
