theorem Th34:
  M is invertible iff Det M <> 0.K
proof
  thus M is invertible implies Det M <> 0.K
  proof
    reconsider N=n as Element of NAT by ORDINAL1:def 12;
    assume M is invertible;
    then consider M1 be Matrix of n,K such that
A1: M is_reverse_of M1 by MATRIX_6:def 3;
    per cases by NAT_1:14;
    suppose
      N=0;
      then Det M=1_K by MATRIXR2:41;
      hence thesis;
    end;
    suppose
A2:   N>=1;
A3:   M*M1=(1.(K,n)) by A1,MATRIX_6:def 2;
      Det (1.(K,n))=1_K by A2,MATRIX_7:16;
      then Det M*Det M1=1_K by A2,A3,MATRIX11:62;
      hence thesis;
    end;
  end;
  set C=(Det M)" * (Matrix_of_Cofactor M)@;
  assume
A4: Det M <> 0.K;
  then
A5: M*C=1.(K,n) by Th30;
  C*M=1.(K,n) by A4,Th33;
  then M is_reverse_of C by A5,MATRIX_6:def 2;
  hence thesis by MATRIX_6:def 3;
end;
