reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a,a1,a2 for Element of K,
  D for non empty set,
  d,d1,d2 for Element of D,
  M,M1,M2 for (Matrix of D),
  A,A1,A2,B1,B2 for (Matrix of K),
  f,g for FinSequence of NAT;
reserve F,F1,F2 for FinSequence_of_Matrix of D,
  G,G9,G1,G2 for FinSequence_of_Matrix of K;
reserve S,S1,S2 for FinSequence_of_Square-Matrix of D,
  R,R1,R2 for FinSequence_of_Square-Matrix of K;

theorem Th46:
  Len S = Width S
proof
  set L=Len S;
  set W=Width S;
A1: dom W=dom S by Def4;
A2: dom L=dom S by Def3;
  now
    let k such that
A3: k in dom L;
    thus L.k = len (S.k) by A3,Def3
      .= width (S.k) by MATRIX_0:24
      .= W.k by A2,A1,A3,Def4;
  end;
  hence thesis by A2,A1,FINSEQ_1:13;
end;
