reserve i,j for Nat;

theorem Th14:
  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 & M1 - M3 = M2 - M3
  holds M1 = 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 and
A5: M1 - M3 = M2 - M3;
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: M3 is Matrix of len M1,width M1,K by A1,A2,A3,A4,A7,MATRIX_0:20;
    M1+-M3+M3=M2+(-M3+M3) by A2,A4,A5,A6,MATRIX_3:3;
    then M1+-M3+M3=M2+(M3+-M3) by A6,MATRIX_3:2;
    then M1+-M3+M3=M2+0.(K,len M1,width M1) by A9,MATRIX_3:5;
    then M1+-M3+M3=M2 by A8,MATRIX_3:4;
    then M1+(-M3+M3)=M2 by A1,A2,A3,A4,A6,MATRIX_3:3;
    then M1+(M3+-M3)=M2 by A6,MATRIX_3:2;
    then
A10: M1+0.(K,len M1,width M1)=M2 by A9,MATRIX_3:5;
    M1 is Matrix of len M1,width M1,K by A7,MATRIX_0:20;
    hence thesis by A10,MATRIX_3:4;
  end;
  suppose
    len M1 = 0;
    hence thesis by A1,CARD_2:64;
  end;
end;
