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 Th60:
  Solutions_of(A,b) is non empty & width A = 0 implies len A=0
proof
  set S=Solutions_of(A,b);
  assume that
A1: S is non empty and
A2: width A=0;
  consider x being object such that
A3: x in S by A1;
  consider f such that
  x=f and
A4: ColVec2Mx f in Solutions_of(A,ColVec2Mx b) by A3;
  consider X such that
  ColVec2Mx f=X and
A5: len X = width A and
  width X = width ColVec2Mx b and
A6: A * X = ColVec2Mx b by A4;
  width (A * X) = width X by A5,MATRIX_3:def 4
    .= 0 by A2,A5,MATRIX_0:def 3;
  hence 0 = len b by A6,MATRIX_0:23
    .= len ColVec2Mx b by MATRIX_0:def 2
    .= len A by A4,Th33;
end;
