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 Th39:
  for M being Matrix of D holds len Mx2FinS(M) = len M * width M
proof
  let M be Matrix of D;
  per cases;
  suppose
A1: len M = 0;
    then Mx2FinS(M) = {} by Def5;
    hence thesis by A1;
  end;
  suppose
A2: len M > 0;
    then consider p being FinSequence of D* such that
A3: Mx2FinS(M) = p.(len M) and
A4: len p = len M and
A5: p.1 = M.1 and
A6: for k st k >= 1 & k < len M holds p.(k+1) = (p.k) ^ M.(k+1) by Def5;
    len M in Seg len M by A2,FINSEQ_1:3;
    then len M in dom p by A4,FINSEQ_1:def 3;
    hence thesis by A3,A4,A5,A6,Th29;
  end;
end;
