let D be set ; :: thesis: for f being FinSequence of D holds
( f is weakly-one-to-one iff for i being Element of NAT st 1 <= i & i < len f holds
f /. i <> f /. (i + 1) )
let f be FinSequence of D; :: thesis: ( f is weakly-one-to-one iff for i being Element of NAT st 1 <= i & i < len f holds
f /. i <> f /. (i + 1) )
thus ( f is weakly-one-to-one implies for i being Element of NAT st 1 <= i & i < len f holds
f /. i <> f /. (i + 1) ) :: thesis: ( ( for i being Element of NAT st 1 <= i & i < len f holds
f /. i <> f /. (i + 1) ) implies f is weakly-one-to-one )

proof

assume A1: f is weakly-one-to-one ; :: 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 A2: ( 1 <= i & i < len f ) ; :: thesis: f /. i <> f /. (i + 1)
A3: f . i = f /. i by A2, FINSEQ_4:24;
( 1 < i + 1 & i + 1 <= len f ) by A2, NAT_1:13;
then f . (i + 1) = f /. (i + 1) by FINSEQ_4:24;
hence f /. i <> f /. (i + 1) by A1, A2, A3, Def2; :: thesis: verum

end;

assume A4: for i being Element of NAT st 1 <= i & i < len f holds
f /. i <> f /. (i + 1) ; :: thesis: f is weakly-one-to-one

now

let i be Element of NAT ; :: thesis: ( 1 <= i & i < len f implies f . i <> f . (i + 1) )
assume A5: ( 1 <= i & i < len f ) ; :: thesis: f . i <> f . (i + 1)
A6: f . i = f /. i by A5, FINSEQ_4:24;
( 1 < i + 1 & i + 1 <= len f ) by A5, NAT_1:13;
then f . (i + 1) = f /. (i + 1) by FINSEQ_4:24;
hence f . i <> f . (i + 1) by A4, A5, A6; :: thesis: verum

end;

hence f is weakly-one-to-one by Def2; :: thesis: verum