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 - (M1 - M2) = M2
proof
  let K be Ring,M1,M2 be Matrix of K;
  assume that
A1: len M1=len M2 and
A2: width M1=width M2;
A3: len (-M1)=len M1 & width (-M1)=width M1 by MATRIX_3:def 2;
A4: len (-M2)=len M2 & width (-M2)=width M2 by MATRIX_3:def 2;
  per cases by NAT_1:3;
  suppose
A5: len M1 > 0;
A6: len (0.(K,len M1,width M1))=len M1 by MATRIX_0:def 2;
    then
A7: width (0.(K,len M1,width M1))=width M1 by A5,MATRIX_0:20;
A8: M2 is Matrix of len M1,width M1,K by A1,A2,A5,MATRIX_0:20;
A9: M1 is Matrix of len M1,width M1,K by A5,MATRIX_0:20;
    M1 - (M1 - M2)=M1+(-M1+--M2) by A1,A2,A4,Th12
      .=M1+(-M1+M2) by Th1
      .=M1+-M1+M2 by A3,MATRIX_3:3
      .=0.(K,len M1,width M1)+M2 by A9,MATRIX_3:5
      .=M2+0.(K,len M1,width M1) by A1,A2,A6,A7,MATRIX_3:2
      .=M2 by A8,MATRIX_3:4;
    hence thesis;
  end;
  suppose
A10: len M1 = 0;
    then len (M1 - (M1 - M2)) = 0 by MATRIX_3:def 3;
    hence thesis by A1,A10,CARD_2:64;
  end;
end;
