let f be FinSequence; :: thesis: ( f is almost-one-to-one implies f is poorly-one-to-one )
assume A1: f is almost-one-to-one ; :: thesis: f is poorly-one-to-one

per cases ( len f <> 2 or len f = 2 ) ;
:: according to JORDAN23:def 3

case A2: len f <> 2 ; :: thesis: for i being Element of NAT st 1 <= i & i < len f holds
f . i <> f . (i + 1)

now

per cases ( len f <> 0 or len f = 0 ) ;

suppose A3: len f <> 0 ; :: thesis: for i being Element of NAT st 1 <= i & i < len f holds
f . i <> f . (i + 1)

let i be Element of NAT ; :: thesis: ( 1 <= i & i < len f implies f . i <> f . (i + 1) )
assume A4: ( 1 <= i & i < len f ) ; :: thesis: f . i <> f . (i + 1)
then A5: i in dom f by FINSEQ_3:27;
A6: i + 1 >= 1 by NAT_1:11;
i + 1 <= len f by A4, NAT_1:13;
then A7: i + 1 in dom f by A6, FINSEQ_3:27;
A8: i <> i + 1 ;
A9: ( not i = 1 or not i + 1 = len f ) by A2;
( not i = len f or not i + 1 = 1 ) by A3;
hence f . i <> f . (i + 1) by A1, A5, A7, A8, A9, Def1; :: thesis: verum

end;

suppose A10: len f = 0 ; :: thesis: for i being Element of NAT st 1 <= i & i < len f holds
f . i <> f . (i + 1)

let i be Element of NAT ; :: thesis: ( 1 <= i & i < len f implies f . i <> f . (i + 1) )
assume ( 1 <= i & i < len f ) ; :: thesis: f . i <> f . (i + 1)
hence f . i <> f . (i + 1) by A10; :: thesis: verum

end;

end;

end;

hence for i being Element of NAT st 1 <= i & i < len f holds
f . i <> f . (i + 1) ; :: thesis: verum

end;

case len f = 2 ; :: thesis: verum

thus verum ; :: thesis: verum

end;

end;