reserve x,y,z for object,
  i,j,n,m for Nat,
  D for non empty set,
  s,t for FinSequence,
  a,a1,a2,b1,b2,d for Element of D,
  p, p1,p2,q,r for FinSequence of D;
reserve M,M1,M2 for Matrix of D;
reserve f for FinSequence of D;

theorem Th38:
  for M being Matrix of D st [i,j] in Indices M & M.i = f holds len
  f = width M & j in dom f
proof
  let M be Matrix of D such that
A1: [i,j] in Indices M;
A2: j in Seg width M by A1,ZFMISC_1:87;
  M is non empty by A1,ZFMISC_1:87;
  then len M > 0;
  then consider p being FinSequence such that
A3: p in rng M and
A4: len p = width M by Def3;
  consider n being Nat such that
A5: for x st x in rng M ex s being FinSequence st s=x & len s
  = n by Def1;
  i in dom M by A1,ZFMISC_1:87;
  then M.i in rng M by FUNCT_1:def 3;
  then
A6: ex s being FinSequence st s = M.i & len s = n by A5;
  ex s being FinSequence st s = p & len s = n by A3,A5;
  hence thesis by A2,A4,A6,FINSEQ_1:def 3;
end;
