reserve i,j for Nat;

theorem Th12:
  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;
  assume
A1: len M1=len M2 & width M1=width M2;
A2: width (-M1)=width M1 by MATRIX_3:def 2;
  then
A3: width (-M1+-M2)=width M1 by MATRIX_3:def 3;
A4: len (M1+M2)=len M1 & width (M1+M2)=width M1 by MATRIX_3:def 3;
A5: len (-M1)=len M1 by MATRIX_3:def 2;
  then
A6: len (-M1+-M2)=len M1 by MATRIX_3:def 3;
A7: len (-M2)=len M2 & width (-M2)=width M2 by MATRIX_3:def 2;
  per cases by NAT_1:3;
  suppose
A8: len M1 > 0;
    then
A9: M2 is Matrix of len M1,width M1,K by A1,MATRIX_0:20;
A10: M1 is Matrix of len M1,width M1,K by A8,MATRIX_0:20;
    M1+M2 +(-M1+-M2) =M1+M2+(-M2+-M1) by A1,A5,A2,A7,MATRIX_3:2
      .=M1+M2+-M2+-M1 by A1,A7,A4,MATRIX_3:3
      .=M1+(M2+-M2)+-M1 by A1,MATRIX_3:3
      .=M1+0.(K,len M1,width M1)+-M1 by A9,MATRIX_3:5
      .=M1+-M1 by A10,MATRIX_3:4
      .=0.(K,len M1,width M1) by A10,MATRIX_3:5;
    hence thesis by A4,A6,A3,Th8;
  end;
  suppose
A11: len M1 = 0;
    then len (-M1) = 0 by MATRIX_3:def 2;
    then
A12: len (-M1+-M2) = 0 by MATRIX_3:def 3;
    len (M1+M2) = 0 by A11,MATRIX_3:def 3;
    then len -(M1+M2) = 0 by MATRIX_3:def 2;
    hence thesis by A12,CARD_2:64;
  end;
end;
