reserve i,j for Nat;

theorem Th18:
  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 (M3 - M1) - (M3 -
  M2) = M2 - M1
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: M3 is Matrix of len M1,width M1,K by A1,A2,A3,A4,MATRIX_0:20;
A7: len (-M2)=len M2 & width (-M2)=width M2 by MATRIX_3:def 2;
A8: width (-M1)=width M1 by MATRIX_3:def 2;
    then
A9: width (-M1+M3)=width M1 by MATRIX_3:def 3;
A10: len (-M3)=len M3 & width (-M3)=width M3 by MATRIX_3:def 2;
A11: len (-M1)=len M1 by MATRIX_3:def 2;
    then
A12: len (-M1+M3)=len M1 by MATRIX_3:def 3;
A13: -M1 is Matrix of len M1,width M1,K by A5,A11,A8,MATRIX_0:20;
    (M3 - M1) - (M3 - M2)=(-M1+M3)-(M3+-M2) by A1,A2,A3,A4,A11,A8,MATRIX_3:2
      .=(-M1+M3)+(-M3+--M2) by A2,A4,A7,Th12
      .=(-M1+M3)+(-M3+M2) by Th1
      .=(-M1+M3)+-M3+M2 by A1,A2,A3,A4,A10,A12,A9,MATRIX_3:3
      .=-M1+(M3+-M3)+M2 by A1,A2,A3,A4,A11,A8,MATRIX_3:3
      .=-M1+(0.(K,len M1,width M1))+M2 by A6,MATRIX_3:5
      .=-M1+M2 by A13,MATRIX_3:4
      .=M2+-M1 by A1,A3,A11,A8,MATRIX_3:2;
    hence thesis;
  end;
  suppose
A14: len M1=0;
A15: len (M2 - M1) = len M2 by MATRIX_3:def 3;
    len ((M3 - M1) - (M3 - M2)) = len (M3 - M1) by MATRIX_3:def 3
      .= len M3 by MATRIX_3:def 3;
    hence thesis by A1,A2,A14,A15,CARD_2:64;
  end;
end;
