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
  n>0 implies M1-M1@ is antisymmetric
proof
  assume
A1: n>0;
  set M2=M1-(M1@);
A2: len M1 = n & width M1=n by MATRIX_0:24;
A3: len (M1@)=n & width (M1@)=n by MATRIX_0:24;
  M2@=M1@+(-(M1@))@ by Th24
    .=M1@+-((M1@)@) by Th27
    .=M1@-M1 by A1,A2,MATRIX_0:57
    .=-(M1-M1@) by A2,A3,MATRIX_4:43;
  hence thesis;
end;
