theorem Th39:
  len b1 > 0 implies width AutMt(f,b1,b2) = len b2
proof
  assume len b1 > 0;
  then len AutMt(f,b1,b2) > 0 by Def8;
  then consider s being FinSequence such that
A1: s in rng AutMt(f,b1,b2) and
A2: len s = width AutMt(f,b1,b2) by MATRIX_0:def 3;
  consider i be Nat such that
A3: i in dom AutMt(f,b1,b2) and
A4: AutMt(f,b1,b2).i = s by A1,FINSEQ_2:10;
  len AutMt(f,b1,b2) = len b1 by Def8;
  then
A5: i in dom b1 by A3,FINSEQ_3:29;
  s = (AutMt(f,b1,b2))/.i by A3,A4,PARTFUN1:def 6
    .= f.(b1/.i) |-- b2 by A5,Def8;
  hence thesis by A2,Def7;
end;
