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 be 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;
