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 Th75:
  for A be Matrix of n,n,K, B be Matrix of K st width A = len B &
  Det A <> 0.K holds the_rank_of (A*B) = the_rank_of B
proof
  let A be Matrix of n,n,K, B be Matrix of K such that
A1: width A = len B and
A2: Det A <>0.K;
  set AB=A*B;
A3: len AB=len A by A1,MATRIX_3:def 4;
A4: len A=n & width A=n by MATRIX_0:24;
A5: width AB=width B by A1,MATRIX_3:def 4;
  per cases;
  suppose
    width AB=0;
    hence thesis by A1,A3,A5,A4,Lm3;
  end;
  suppose
A6: width AB>0;
    then Space_of_Solutions_of B = Space_of_Solutions_of AB & dim
    Space_of_Solutions_of B = width B-the_rank_of B by A1,A2,A5,Th68,Th74;
    then width B-the_rank_of B=width B-the_rank_of AB by A5,A6,Th68;
    hence thesis;
  end;
end;
