reserve

  k,n,m,i,j for Element of NAT,
  K for Field;
reserve L for non empty addLoopStr;
reserve G for non empty multLoopStr;

theorem Th19:
  for A being Matrix of n,K holds A is invertible iff ex B being
  Matrix of n,K st B*A=1.(K,n) & A*B=1.(K,n)
proof
  let A be Matrix of n,K;
  thus A is invertible implies ex B being Matrix of n,K st B*A=1.(K,n) & A*B=
  1.(K,n)
  proof
    assume A is invertible;
    then (A~)*A=1.(K,n) & A*(A~)=1.(K,n) by Th18;
    hence thesis;
  end;
  thus thesis by Th18;
end;
