reserve x,y for set,
  i,j,k,l,m,n for Nat,
  K for Field,
  N for without_zero finite Subset of NAT,
  a,b for Element of K,
  A,B,B1,B2,X,X1,X2 for (Matrix of K),
  A9 for (Matrix of m,n,K),
  B9 for (Matrix of m,k,K);
reserve D for non empty set,
  bD for FinSequence of D,
  b,f,g for FinSequence of K,
  MD for Matrix of D;

theorem Th56:
  Solutions_of(0.(K,n,0),0.(K,n,0)) = {{}}
proof
  per cases;
  suppose
    n=0;
    hence thesis by Th51;
  end;
  suppose
A1: n>0;
    set B=0.(K,n,0);
    set A=0.(K,n,0);
    reconsider E={} as Matrix of 0,0,K by MATRIX_0:13;
A2: width A=0 by A1,MATRIX_0:23;
    then
A3: for i,j st [i,j] in Indices B holds B*(i,j)=(A*E)*(i,j) by ZFMISC_1:90;
A4: Solutions_of(A,B) c= {{}}
    proof
      let x be object;
      assume x in Solutions_of(A,B);
      then reconsider X=x as Matrix of 0,0,K by A1,Th53;
      len X=0 by MATRIX_0:def 2;
      then X={};
      hence thesis by TARSKI:def 1;
    end;
A5: len E=0;
A6: width E=0 by MATRIX_0:24;
    then
A7: width (A*E)=0 by A2,A5,MATRIX_3:def 4;
A8: len A=n by A1,MATRIX_0:23;
    then len (A*E)=n by A2,A5,MATRIX_3:def 4;
    then A*E=B by A8,A2,A7,A3,MATRIX_0:21;
    then E in Solutions_of(A,B) by A2,A5,A6;
    hence thesis by A4,ZFMISC_1:33;
  end;
end;
