reserve i,j,n for Nat,
  K for Field,
  a for Element of K,
  M,M1,M2,M3,M4 for Matrix of n,K;
reserve A for Matrix of K;

theorem Th27:
  for K being Field,n being Nat,M1 being Matrix of n,K holds (-M1) @=-(M1@)
proof
  let K,n,M1;
  for i,j st [i,j] in Indices ((-M1)@) holds ((-M1)@)*(i,j)=(-(M1@))*(i,j)
  proof
    let i,j;
    assume
A1: [i,j] in Indices ((-M1)@);
    then
A2: [i,j] in Indices M1@ by MATRIX_0:26;
    [i,j] in [:Seg n, Seg n:] by A1,MATRIX_0:24;
    then i in Seg n & j in Seg n by ZFMISC_1:87;
    then
A3: [j,i] in [:Seg n, Seg n:] by ZFMISC_1:87;
    then
A4: [j,i] in Indices M1 by MATRIX_0:24;
    [j,i] in Indices (-M1) by A3,MATRIX_0:24;
    then ((-M1)@)*(i,j)=(-M1)*(j,i) by MATRIX_0:def 6
      .=-(M1*(j,i)) by A4,MATRIX_3:def 2
      .=-(M1@*(i,j)) by A4,MATRIX_0:def 6
      .=(-(M1@))*(i,j) by A2,MATRIX_3:def 2;
    hence thesis;
  end;
  hence thesis by MATRIX_0:27;
end;
