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 Th51:
  for A be Matrix of 0,n,K, B be Matrix of 0,m,K holds Solutions_of(A,B) = {{}}
proof
  let A be Matrix of 0,n,K, B be Matrix of 0,m,K;
A1: len A=0 by MATRIX_0:def 2;
A2: Solutions_of(A,B) c= {{}}
  proof
    let x be object;
    assume x in Solutions_of(A,B);
    then ex X st X=x &len X = width A & width X = width B & A * X = B;
    then x={} by A1,MATRIX_0:def 3;
    hence thesis by TARSKI:def 1;
  end;
  len B=0 by MATRIX_0:def 2;
  then
A3: B={} & width B=0 by MATRIX_0:def 3;
A4: width A=0 by A1,MATRIX_0:def 3;
  then len (A*A)=0 by A1,MATRIX_3:def 4;
  then A*A={};
  then A in Solutions_of(A,B) by A1,A4,A3;
  hence thesis by A2,ZFMISC_1:33;
end;
