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 - M3 = M4 - M2 holds M1 + M2 = M3 + 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 - M3 = M4 - M2;
A8: len (-M3)=len M1 & width (-M3)=width M1 by A1,A2,A4,A5,MATRIX_3:def 2;
A9: len (-M2)=len M1 & width (-M2)=width M1 by A1,A4,MATRIX_3:def 2;
  then M1+-M3=-M2+M4 by A1,A2,A3,A4,A5,A6,A7,MATRIX_3:2;
  then M1--M2=M4--M3 by A1,A2,A3,A4,A5,A6,A9,A8,Th28;
  then M1+M2=M4--M3 by Th1;
  then M1+M2=M4+M3 by Th1;
  hence thesis by A3,A6,MATRIX_3:2;
end;
