reserve i,j for Nat;

theorem
  for K being Ring,M1,M2,M3,M4 being Matrix of K st len M1=len M2 & len
M2=len M3 & len M3=len M4 & width M1=width M2 & width M2 = width M3 & width M3=
  width M4 & M1 - M2 = M3 - M4 holds M1 - M3 = M2 - M4
proof
  let K be Ring,M1,M2,M3,M4 be Matrix of K;
  assume that
A1: len M1=len M2 and
A2: len M2=len M3 and
A3: len M3=len M4 and
A4: width M1=width M2 and
A5: width M2 = width M3 and
A6: width M3=width M4 and
A7: M1 - M2 = M3 - M4;
A8: len (-M4)=len M1 & width (-M4)=width M1 by A1,A2,A3,A4,A5,A6,MATRIX_3:def 2
;
A9: len (M3+-M4)=len M1 & width (M3+-M4)=width M1 by A1,A2,A4,A5,MATRIX_3:def 3
;
A10: len (-M2)=len M2 & width (-M2)=width M2 by MATRIX_3:def 2;
A11: len (M1+-M3)=len M1 & width (M1+-M3)=width M1 by MATRIX_3:def 3;
A12: len (M1+-M2)=len M1 & width (M1+-M2)=width M1 by MATRIX_3:def 3;
A13: len (M1+-M3)=len M1 & width (M1+-M3)=width M1 by MATRIX_3:def 3;
A14: len (M2+-M4)=len M1 & width (M2+-M4)=width M1 by A1,A4,MATRIX_3:def 3;
A15: len (-M3)=len M3 & width (-M3)=width M3 by MATRIX_3:def 2;
  per cases by NAT_1:3;
  suppose
    len M1 > 0;
    then M3+-M4 is Matrix of len M1,width M1,K by A9,MATRIX_0:20;
    then M1+-M2+-(M3 +- M4)=0.(K,len M1,width M1) by A7,MATRIX_3:5;
    then M1+-M2+(-M3 +-- M4)=0.(K,len M1,width M1) by A1,A2,A4,A5,A8,Th12;
    then M1+-M2+(-M3 + M4)=0.(K,len M1,width M1) by Th1;
    then M1+-M2+-M3 + M4=0.(K,len M1,width M1) by A1,A2,A4,A5,A15,A12,
MATRIX_3:3;
    then M1+(-M2+-M3) + M4=0.(K,len M1,width M1) by A1,A4,A10,MATRIX_3:3;
    then M1+(-M3+-M2) + M4=0.(K,len M1,width M1) by A2,A5,A10,A15,MATRIX_3:2;
    then M1+-M3+-M2 + M4=0.(K,len M1,width M1) by A1,A2,A4,A5,A15,MATRIX_3:3;
    then M1+-M3+(-M2 + M4)=0.(K,len M1,width M1) by A1,A4,A10,A11,MATRIX_3:3;
    then M1+-M3+(-M2 + --M4)=0.(K,len M1,width M1) by Th1;
    then M1+-M3-(M2 + -M4)=0.(K,len M1,width M1) by A1,A4,A8,Th12;
    hence thesis by A13,A14,Th7;
  end;
  suppose
    len M1 = 0;
    then len (M1 - M3) = 0 & len (M2 - M4) = 0 by A1,MATRIX_3:def 3;
    hence thesis by CARD_2:64;
  end;
end;
