reserve i,j for Nat;

theorem Th15:
  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 & M3 - M1 = M3 - M2
  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: M3 - M1 = M3 - M2;
  per cases by NAT_1:3;
  suppose
A6: len M1>0;
    then
A7: M3 is Matrix of len M1,width M1,K by A1,A2,A3,A4,MATRIX_0:20;
A8: len (-M2)=len M2 & width (-M2)=width M2 by MATRIX_3:def 2;
    then
A9: -M2 is Matrix of len M1,width M1,K by A1,A3,A6,MATRIX_0:20;
A10: len (-M1)=len M1 & width (-M1)=width M1 by MATRIX_3:def 2;
    then -M1+M3=M3+-M2 by A1,A2,A3,A4,A5,MATRIX_3:2;
    then -M1+M3=-M2+M3 by A2,A4,A8,MATRIX_3:2;
    then -M1+M3+-M3=-M2+(M3+-M3) by A2,A4,A8,MATRIX_3:3;
    then -M1+M3+-M3=-M2+0.(K,len M1,width M1) by A7,MATRIX_3:5;
    then -M1+M3+-M3=-M2 by A9,MATRIX_3:4;
    then -M1+(M3+-M3)=-M2 by A1,A2,A3,A4,A10,MATRIX_3:3;
    then
A11: -M1+0.(K,len M1,width M1)=-M2 by A7,MATRIX_3:5;
    -M1 is Matrix of len M1,width M1,K by A6,A10,MATRIX_0:20;
    then -M1=-M2 by A11,MATRIX_3:4;
    then --M1=M2 by Th1;
    hence thesis by Th1;
  end;
  suppose
    len M1=0;
    hence thesis by A1,CARD_2:64;
  end;
end;
