 reserve X,Y for set,
         n,m,k,i for Nat,
         r for Real,
         R for Element of F_Real,
         K for Field,
         f,f1,f2,g1,g2 for FinSequence,
         rf,rf1,rf2 for real-valued FinSequence,
         cf,cf1,cf2 for complex-valued FinSequence,
         F for Function;

theorem Th16:
  for A be Matrix of n,m,K st the_rank_of A = n
    ex B be Matrix of m-' n,m,K st the_rank_of(A^B) = m
proof
  let A be Matrix of n,m,K such that
   A1: the_rank_of A=n;
  per cases;
  suppose A2: n=0;
   then m-' n=m-n by XREAL_0:def 2;
   then reconsider ONE=1.(K,m) as Matrix of m-' n,m,K by A2;
   Det 1.(K,m)<>0.K by LAPLACE:34;
   then A3: the_rank_of ONE=m by MATRIX13:83;
   len A=0 by A2,MATRIX_0:def 2;
   then A is empty;
   then A^ONE=ONE by FINSEQ_1:34;
   hence thesis by A3;
  end;
  suppose n>0;
   hence thesis by A1,Lm1;
  end;
end;
