reserve i,j for Nat;

theorem Th1:
  for K being Ring, M being Matrix of K holds --M=M
proof
  let K be Ring, M be Matrix of K;
  per cases by NAT_1:3;
  suppose
A1: len M = 0;
    then len -M = 0 by MATRIX_3:def 2;
    then len --M = 0 by MATRIX_3:def 2;
    hence thesis by A1,CARD_2:64;
  end;
  suppose
A2: len M>0;
    len (-M) = len M & width (-M) = width M by MATRIX_3:def 2;
    then
A3: -M is Matrix of len M,width M,K by A2,MATRIX_0:20;
    M is Matrix of len M,width M,K by A2,MATRIX_0:20;
    then
A4: Indices (-M)=Indices M by A3,MATRIX_0:26;
A5: for i,j st [i,j] in Indices M holds M*(i,j) = (--M)*(i,j)
    proof
      let i,j;
      assume
A6:   [i,j] in Indices M;
      then (-M)*(i,j)= -(M*(i,j)) by MATRIX_3:def 2;
      then (-(-M))*(i,j) = --(M*(i,j)) by A4,A6,MATRIX_3:def 2;
      hence thesis by RLVECT_1:17;
    end;
    width (--M) = width (-M) by MATRIX_3:def 2;
    then
A7: width (--M)= width M by MATRIX_3:def 2;
    len (--M) = len (-M) by MATRIX_3:def 2;
    then len (--M) =len M by MATRIX_3:def 2;
    hence thesis by A7,A5,MATRIX_0:21;
  end;
end;
