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;

theorem
  for M being Matrix of n,m,D st [i,j] in Indices M holds 1 <= i & i <=
  n & 1 <= j & j <= m
proof
  let M be Matrix of n,m,D such that
A1: [i,j] in Indices M;
A2: len M = n by Def2;
  per cases;
  suppose
A3: n = 0;
A4: i in dom M by A1,ZFMISC_1:87;
    then
A5: 1 <= i by FINSEQ_3:25;
    i <= 0 by A2,A3,A4,FINSEQ_3:25;
    hence thesis by A5;
  end;
  suppose
    n > 0;
    then m = width M by Th23;
    hence thesis by A1,A2,Th32;
  end;
end;
