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 Th28:
  AutMt(id V1,b1,b1) = 1.(K,len b1)
proof
  set A=AutMt(id V1,b1,b1);
  set ONE=1.(K,len b1);
A1: len A=len b1 by MATRIX_0:def 2;
A2: now
    let i such that
A3: 1<=i & i<=len b1;
A4: i in dom b1 by A3,FINSEQ_3:25;
A5: i in Seg len b1 by A3;
    i in dom A by A1,A3,FINSEQ_3:25;
    hence A.i = A/.i by PARTFUN1:def 6
      .= ((id V1).(b1/.i)) |-- b1 by A4,MATRLIN:def 8
      .= (b1/.i) |-- b1
      .= Line(ONE,i) by A4,Th19
      .= ONE.i by A5,MATRIX_0:52;
  end;
  len ONE=len b1 by MATRIX_0:def 2;
  hence thesis by A1,A2;
end;
