reserve i,j for Nat;

theorem Th28:
  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 = M4 - M2
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 (-M2)=len M1 & width (-M2)=width M1 by A1,A4,MATRIX_3:def 2;
  M1+M2=M4+M3 by A3,A6,A7,MATRIX_3:2;
  then M1+M2+-M2=M4+(M3+-M2) by A3,A6,MATRIX_3:3;
  then M1+M2+-M2=M4+(-M2+M3) by A1,A2,A4,A5,A8,MATRIX_3:2;
  then M1+(M2-M2)=M4+(-M2+M3) by A1,A4,MATRIX_3:3;
  then M1=M4+(-M2+M3) by A1,A4,Th20;
  then
A9: M1=M4+-M2+M3 by A1,A2,A3,A4,A5,A6,A8,MATRIX_3:3;
  len (M4+-M2)=len M1 & width (M4+-M2)=width M1 by A1,A2,A3,A4,A5,A6,
MATRIX_3:def 3;
  hence thesis by A1,A2,A4,A5,A9,Th21;
end;
