reserve i,n for Nat,
  K for Field,
  M1,M2,M3,M4 for Matrix of n,K;

theorem Th55:
  Trace (0.(K,n))=0.K
proof
  len (diagonal_of_Matrix 0.(K,n))=n by MATRIX_3:def 10;
  then
A1: dom (diagonal_of_Matrix 0.(K,n))= Seg n by FINSEQ_1:def 3;
  for i st i in dom (diagonal_of_Matrix 0.(K,n)) holds (diagonal_of_Matrix
  0.(K,n))/.i=0.K
  proof
    let i;
    assume
A2: i in dom (diagonal_of_Matrix 0.(K,n));
    Indices (0.(K,n))=[:Seg n, Seg n:] by MATRIX_0:24;
    then [i,i] in Indices (0.(K,n,n)) by A1,A2,ZFMISC_1:87;
    then
A3: (0.(K,n))*(i,i)=0.K by MATRIX_3:1;
    (diagonal_of_Matrix 0.(K,n)).i=(0.(K,n))*(i,i) by A1,A2,MATRIX_3:def 10;
    hence thesis by A2,A3,PARTFUN1:def 6;
  end;
  hence thesis by MATRLIN:11;
end;
