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);
reserve p,p1 for FinSequence of K;

theorem
  Len mlt(p,G) = Len G & Width mlt(p,G) = Width G
proof
  set M=mlt(p,G);
A1: dom Len M=dom M by Def3;
A2: dom M=dom G by Def9;
A3: dom Len G=dom G by Def3;
  now
    let i such that
A4: i in dom Len M;
    thus (Len M).i = len (M.i) by A4,Def3
      .= len (p/.i*G.i) by A1,A4,Def9
      .= len (G.i) by MATRIX_3:def 5
      .= (Len G).i by A1,A3,A2,A4,Def3;
  end;
  hence Len M=Len G by A1,A3,A2,FINSEQ_1:13;
A5: dom Width M=dom M by Def4;
A6: dom Width G=dom G by Def4;
  now
    let i such that
A7: i in dom Width M;
    thus (Width M).i = width (M.i) by A7,Def4
      .= width (p/.i*G.i) by A5,A7,Def9
      .= width (G.i) by MATRIX_3:def 5
      .= (Width G).i by A6,A5,A2,A7,Def4;
  end;
  hence thesis by A6,A5,A2,FINSEQ_1:13;
end;
