reserve i,j for Nat;

theorem Th7:
  for K being Ring,M1,M2 being Matrix of K st len M1=len M2 &
  width M1=width M2 & M1 - M2 = 0.(K,len M1,width M1) holds M1 = M2
proof
  let K be Ring,M1,M2 be Matrix of K;
  assume that
A1: len M1=len M2 and
A2: width M1=width M2 and
A3: M1 - M2 = 0.(K,len M1,width M1);
  per cases by NAT_1:3;
  suppose
A4: len M1 > 0;
    then
A5: M2 is Matrix of len M1,width M1,K by A1,A2,MATRIX_0:20;
A6: len (-M2)=len M2 & width (-M2)=width (M2) by MATRIX_3:def 2;
A7: len (0.(K,len M1,width M1))=len M1 by MATRIX_0:def 2;
    then width (0.(K,len M1,width M1))=width M1 by A4,MATRIX_0:20;
    then M1+(-M2)+M2 = M2+0.(K,len M1,width M1) by A1,A2,A3,A7,MATRIX_3:2
      .=M2 by A5,MATRIX_3:4;
    then M1+((-M2)+M2)=M2 by A1,A2,A6,MATRIX_3:3;
    then M1+(M2+(-M2))=M2 by A6,MATRIX_3:2;
    then
A8: M1+(0.(K,len M1,width M1))=M2 by A5,MATRIX_3:5;
    M1 is Matrix of len M1,width M1,K by A4,MATRIX_0:20;
    hence thesis by A8,MATRIX_3:4;
  end;
  suppose
    len M1 = 0;
    hence thesis by A1,CARD_2:64;
  end;
end;
