reserve i,j for Nat;

theorem
  for K being Ring,M1,M2 being Matrix of K st len M1=len M2 & width M1=
  width M2 holds M1 = - M2 - (- M1 - M2)
proof
  let K be Ring,M1,M2 be Matrix of K;
A1: len (M1+M2)=len M1 & width (M1+M2)=width M1 by MATRIX_3:def 3;
  assume
A2: len M1=len M2 & width M1=width M2;
  then
A3: len (-M2)=len M1 & width (-M2)=width M1 by MATRIX_3:def 2;
  len (-M1)=len M1 & width (-M1)=width M1 by MATRIX_3:def 2;
  then -M2-(-M1-M2)=-M2+(--M1+--M2) by A3,Th12
    .=-M2+(M1+--M2) by Th1
    .=-M2+(M1+M2) by Th1
    .=M1+M2+-M2 by A3,A1,MATRIX_3:2;
  hence thesis by A2,Th38;
end;
