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 holds M1 - M3 = (M1 - M2) - (
  M3 - 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;
A5: len (-M2)=len M2 & width (-M2)=width M2 by MATRIX_3:def 2;
A6: len (-M3)=len M3 & width (-M3)=width M3 by MATRIX_3:def 2;
  per cases by NAT_1:3;
  suppose
A7: len M1>0;
    then
A8: M2 is Matrix of len M1,width M1,K by A1,A3,MATRIX_0:20;
A9: len (M1+-M2)=len M1 & width (M1+-M2)=width M1 by MATRIX_3:def 3;
A10: M1 is Matrix of len M1,width M1,K by A7,MATRIX_0:20;
    (M1 - M2) - (M3 - M2)=M1+-M2+(-M3+--M2) by A2,A4,A5,Th12
      .=M1+-M2+(-M3+M2) by Th1
      .=M1+-M2+(M2+-M3) by A2,A4,A6,MATRIX_3:2
      .=M1+-M2+M2+-M3 by A1,A3,A9,MATRIX_3:3
      .=M1+(-M2+M2)+-M3 by A1,A3,A5,MATRIX_3:3
      .=M1+(M2+-M2)+-M3 by A5,MATRIX_3:2
      .=M1+0.(K,len M1,width M1)+-M3 by A8,MATRIX_3:5
      .=M1+-M3 by A10,MATRIX_3:4;
    hence thesis;
  end;
  suppose
A11: len M1 = 0;
    then len (M1-M2) = 0 by MATRIX_3:def 3;
    then
A12: len ((M1 - M2) - (M3 - M2)) = 0 by MATRIX_3:def 3;
    len (M1-M3) = 0 by A11,MATRIX_3:def 3;
    hence thesis by A12,CARD_2:64;
  end;
end;
