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 Th18:
  for A,B being Matrix of n,K holds A is invertible & B=A~ iff B*A
  =1.(K,n) & A*B=1.(K,n)
proof
  let A,B be Matrix of n,K;
  hereby
    assume A is invertible & B=A~;
    then B is_reverse_of A by MATRIX_6:def 4;
    hence B*A=1.(K,n) & A*B=1.(K,n) by MATRIX_6:def 2;
  end;
  hereby
    assume B*A=1.(K,n) & A*B=1.(K,n);
    then
A1: B is_reverse_of A by MATRIX_6:def 2;
    hence A is invertible by MATRIX_6:def 3;
    hence B=A~ by A1,MATRIX_6:def 4;
  end;
end;
