reserve
  j, k, l, n, m, t,i for Nat,
  K for comRing, 
  a for Element of K,
  M,M1,M2 for Matrix of n,m,K,
  pK,qK for FinSequence of K,
  A for Matrix of n,K;

theorem Th14:
  a is left_invertible & a is right_mult-cancelable &
  l in dom (1.(K,n)) implies SXLine((1.(K,n)),l,a) is
  invertible & (SXLine((1.(K,n)),l,a))~ = SXLine((1.(K,n)),l, / a)
proof
  assume that
A0: a is left_invertible & a is right_mult-cancelable and
A1: l in dom (1.(K,n));
  set A = SXLine((1.(K,n)),l,a);
A3: dom A = Seg len A by FINSEQ_1:def 3
    .= Seg len (1.(K,n)) by Def2
    .= dom (1.(K,n)) by FINSEQ_1:def 3;
  set B = SXLine((1.(K,n)),l,/ a);
A4: dom B = Seg len B by FINSEQ_1:def 3
    .= Seg len (1.(K,n)) by Def2
    .= dom (1.(K,n)) by FINSEQ_1:def 3;
A5: width B = width (1.(K,n)) by Th1;
  for i,j st [i,j] in Indices (1.(K,n)) holds (1.(K,n))*(i,j) = SXLine(B,l
  ,a)*(i,j)
  proof
    let i,j;
    assume
A6: [i,j] in Indices (1.(K,n));
    then
A7: i in dom (1.(K,n)) by ZFMISC_1:87;
A8: j in Seg width (1.(K,n)) by A6,ZFMISC_1:87;
    then
A9: i <> l implies B*(i,j) = (1.(K,n))*(i,j) by A7,Def2;
    B*(l,j) = (/ a)*((1.(K,n))*(l,j)) by A1,A8,Def2;
    then
A10: SXLine(B,l,a)*(l,j) = a*((/ a)*((1.(K,n))*(l,j))) by A1,A5,A4,A8,Def2
      .= (a*(/ a))*((1.(K,n))*(l,j)) by GROUP_1:def 3
      .= ((/ a)*a)*((1.(K,n))*(l,j))
      .= (1_K)*((1.(K,n))*(l,j)) by A0, ALGSTR_0:def 30
      .= (1.(K,n))*(l,j);
    i <> l implies SXLine(B,l,a)*(i,j) = B*(i,j) by A5,A4,A7,A8,Def2;
    hence thesis by A9,A10;
  end;
  then
A11: 1.(K,n) = SXLine(B,l,a) by MATRIX_0:27;
A12: width A = width (1.(K,n)) by Th1;
  for i,j st [i,j] in Indices (1.(K,n)) holds (1.(K,n))*(i,j) = SXLine(A,
  l,/ a)*(i,j)
  proof
    let i,j;
    assume
A13: [i,j] in Indices (1.(K,n));
    then
A14: i in dom (1.(K,n)) by ZFMISC_1:87;
A15: j in Seg width (1.(K,n)) by A13,ZFMISC_1:87;
    then
A16: i <> l implies A*(i,j) = (1.(K,n))*(i,j) by A14,Def2;
    A*(l,j) = a*((1.(K,n))*(l,j)) by A1,A15,Def2;
    then
A17: SXLine(A,l,/a)*(l,j) = (/a)*(a*((1.(K,n))*(l,j))) by A1,A12,A3,A15,Def2
      .= ((/a)*a)*((1.(K,n))*(l,j)) by GROUP_1:def 3
      .= (1_K)*((1.(K,n))*(l,j)) by A0,ALGSTR_0:def 30
      .= (1.(K,n))*(l,j);
    i <> l implies SXLine(A,l,/a)*(i,j) = A*(i,j) by A12,A3,A14,A15,Def2;
    hence thesis by A16,A17;
  end;
  then
A18: 1.(K,n) = SXLine(A,l,/a) by MATRIX_0:27;
  A * B = SXLine(B,l,a) & B * A = SXLine(A,l,/a) by A1,Th7;
  then
A19: B is_reverse_of A by A11,A18,MATRIX_6:def 2;
  then A is invertible by MATRIX_6:def 3;
  hence thesis by A19,MATRIX_6:def 4;
end;
