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 & M2 - M1 = M2 holds M1=0.(K,len M1,width 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: M2 - M1 = M2;
  per cases by NAT_1:3;
  suppose
A3: len M1 > 0;
A4: len (-M1)=len M1 & width (-M1)=width (M1) by MATRIX_3:def 2;
A5: M2 is Matrix of len M1, width M1, K by A1,A3,MATRIX_0:20;
    then M2+(-M1)+(-M2) = 0.(K,len M1,width M1) by A2,MATRIX_3:5;
    then (-M1)+M2+(-M2) = 0.(K,len M1,width M1) by A1,A4,MATRIX_3:2;
    then (-M1)+(M2+(-M2)) = 0.(K,len M1,width M1) by A1,A4,MATRIX_3:3;
    then
A6: -M1+ (0.(K,len M1,width M1))=0.(K,len M1,width M1) by A5,MATRIX_3:5;
    -M1 is Matrix of len M1,width M1,K by A3,A4,MATRIX_0:20;
    then -M1= 0.(K,len M1,width M1) by A6,MATRIX_3:4;
    then M1=- 0.(K,len M1,width M1) by Th1;
    hence thesis by Th9;
  end;
  suppose
A7: len M1 = 0;
    then len (0.(K,len M1,width M1)) = 0;
    hence thesis by A7,CARD_2:64;
  end;
end;
