reserve i,j for Nat;

theorem Th8:
  for K being Ring,M1,M2 being Matrix of K st len M1=len M2 &
  width M1=width M2 & M1+M2= 0.(K,len M1,width M1) holds M2=-M1
proof
  let K be Ring,M1,M2 be Matrix of K;
  assume that
A1: len M1=len M2 & width M1=width M2 and
A2: M1+M2= 0.(K,len M1,width M1);
A3: len (-M2)=len M2 & width (-M2)=width M2 by MATRIX_3:def 2;
  M1-(-M2)= 0.(K,len M1,width M1) by A2,Th1;
  then M1=-M2 by A1,A3,Th7;
  hence thesis by Th1;
end;
