reserve i, j, m, n, k for Nat,
  x, y for set,
  K for Field,
  a,a1 for Element of K;
reserve V1,V2,V3 for finite-dimensional VectSp of K,
  f for Function of V1,V2,

  b1,b19 for OrdBasis of V1,
  B1 for FinSequence of V1,
  b2 for OrdBasis of V2,
  B2 for FinSequence of V2,

  B3 for FinSequence of V3,
  v1,w1 for Element of V1,
  R,R1,R2 for FinSequence of V1,
  p,p1,p2 for FinSequence of K;

theorem
  AutEqMt(id V1,b1,b19) is invertible & AutEqMt(id V1,b19,b1) = AutEqMt(
  id V1,b1,b19)~
proof
  set A = AutEqMt(id V1,b1,b19);
A1: 1_K<>0.K;
A2: len b1 = dim V1 by Th21
    .= len b19 by Th21;
  then reconsider A9= AutEqMt(id V1,b19,b1) as Matrix of len b1,len b1,K;
A3: A=AutMt(id V1,b1,b19) & A9=AutMt(id V1,b19,b1) by A2,Def2;
  per cases;
  suppose
    len b1=0;
    then Det A=1_K & A9=A~ by MATRIXR2:41,MATRIX_0:45;
    hence thesis by A1,LAPLACE:34;
  end;
  suppose
A4: len b1+0>0;
    dom id V1=the carrier of V1;
    then
A5: (id V1) * (id V1) = id V1 by RELAT_1:52;
    len b1=dim V1 by Th21;
    then len b1 = len b19 by Th21;
    then
A6: A * A9 = AutMt((id V1)*(id V1),b1,b1) by A3,A4,MATRLIN:41
      .= 1.(K,len b1) by A5,Th28;
    len b1>=1 by A4,NAT_1:19;
    then 1_K = Det (A * A9) by A6,MATRIX_7:16
      .= Det A * Det A9 by A4,MATRIX11:62;
    then Det A <> 0.K;
    then
A7: A is invertible by LAPLACE:34;
    then A~ is_reverse_of A by MATRIX_6:def 4;
    then A * (A~)=1.(K,len b1) by MATRIX_6:def 2;
    hence thesis by A6,A7,MATRIX_8:19;
  end;
end;
