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
  for K being Field,n being Nat,M1,M2 being Matrix of n,K st 
  M1 is antisymmetric & M2 is antisymmetric holds 
    M1-M2 is antisymmetric
proof
  let K,n,M1,M2;
  assume that
A1: M1 is antisymmetric and
A2: M2 is antisymmetric;
A3: len (-M2)=n & width (-M2)=n by MATRIX_0:24;
A4: len M1=n & width M1=n by MATRIX_0:24;
  (M1-M2)@=M1@+(-M2)@ by Th24
    .=-M1+(-M2)@ by A1
    .=-M1+-(M2@) by Th27
    .=-M1+-(-M2) by A2
    .=-(M1-M2) by A3,A4,MATRIX_4:12;
  hence thesis;
end;
