reserve i,j for Nat;

theorem Th26:
  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 holds M1 + M3 = (M1 +
  M2) - (M2 - M3)
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;
  per cases by NAT_1:3;
  suppose
A5: len M1 > 0;
    then
A6: M2 is Matrix of len M1,width M1,K by A1,A3,MATRIX_0:20;
A7: len (-M2)=len M1 & width (-M2)=width M1 by A1,A3,MATRIX_3:def 2;
A8: len (-M3)=len M1 & width (-M3)=width M1 by A1,A2,A3,A4,MATRIX_3:def 2;
A9: len (M1+M2)=len M1 & width (M1+M2)=width M1 by MATRIX_3:def 3;
    M1 is Matrix of len M1,width M1,K by A5,MATRIX_0:20;
    hence M1 + M3 = M1+0.(K,len M1,width M1)+M3 by MATRIX_3:4
      .=M1+(M2+-M2)+M3 by A6,MATRIX_3:5
      .=M1+M2+-M2+M3 by A1,A3,MATRIX_3:3
      .=M1+M2+(-M2+M3) by A7,A9,MATRIX_3:3
      .=M1+M2+(-M2+--M3) by Th1
      .=(M1 + M2) - (M2 - M3) by A1,A3,A8,Th12;
  end;
  suppose
A10: len M1 = 0;
A11: len ((M1 + M2) - (M2 - M3)) = len (M1 + M2) by MATRIX_3:def 3
      .= len M1 by MATRIX_3:def 3;
    len (M1 + M3) = len M1 by MATRIX_3:def 3;
    hence thesis by A10,A11,CARD_2:64;
  end;
end;
