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;
reserve N for (Matrix of n,K),
  N1 for (Matrix of m,K);

theorem
  Len 1.(K,f) = f & Width 1.(K,f) = f
proof
  set ONE=1.(K,f);
A1: dom ONE=dom f by Def8;
A2: dom Len ONE=dom ONE by Def3;
  now
    let i such that
A3: i in dom Len ONE;
    thus (Len ONE).i = len (ONE.i) by A3,Def3
      .= len (1.(K,f.i)) by A2,A3,Def8
      .= f.i by MATRIX_0:24;
  end;
  hence Len ONE=f by A2,A1,FINSEQ_1:13;
A4: dom Width ONE=dom ONE by Def4;
  now
    let i such that
A5: i in dom Width ONE;
    thus (Width ONE).i = width (ONE.i) by A5,Def4
      .= width (1.(K,f.i)) by A4,A5,Def8
      .= f.i by MATRIX_0:24;
  end;
  hence thesis by A4,A1,FINSEQ_1:13;
end;
