reserve i,j for Nat;

theorem
  for K being Ring,M1,M2,M3 being Matrix of K st len M1=len M2 & len M2
  =len M3 & width M1=width M2 & width M2=width M3 & M1+M3=M2+M3 holds M1=M2
proof
  let K be Ring,M1,M2,M3 be Matrix of K;
  assume that
A1: len M1=len M2 and
A2: len M2=len M3 and
A3: width M1=width M2 and
A4: width M2=width M3 and
A5: M1+M3=M2+M3;
A6: M3+-M3=0.(K,len M1,width M1) by A1,A2,A3,A4,Th2;
  M1+(M3+(-M3))=M2+M3+(-M3) by A1,A2,A3,A4,A5,MATRIX_3:3;
  then
A7: M1+(M3+-M3)=M2+(M3+(-M3)) by A2,A4,MATRIX_3:3;
  per cases by NAT_1:3;
  suppose
A8: len M1 > 0;
    then M2 is Matrix of len M1,width M1,K by A1,A3,MATRIX_0:20;
    then
A9: M2+0.(K,len M1,width M1)=M2 by MATRIX_3:4;
    M1 is Matrix of len M1,width M1,K by A8,MATRIX_0:20;
    hence thesis by A7,A6,A9,MATRIX_3:4;
  end;
  suppose
    len M1 = 0;
    hence thesis by A1,CARD_2:64;
  end;
end;
