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;
