let f be FinSequence of (TOP-REAL 2); :: thesis: for n being Element of NAT st f is special holds
f /^ n is special
let n be Element of NAT ; :: thesis: ( f is special implies f /^ n is special )
assume A1: f is special ; :: thesis: f /^ n is special

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

suppose A2: n <= len f ; :: thesis: f /^ n is special

let i be Nat; :: according to TOPREAL1:def 7 :: thesis: ( not 1 <= i or not i + 1 <= len (f /^ n) or ((f /^ n) /. i) `1 = ((f /^ n) /. (i + 1)) `1 or ((f /^ n) /. i) `2 = ((f /^ n) /. (i + 1)) `2 )
assume that
A3: 1 <= i and
A4: i + 1 <= len (f /^ n) ; :: thesis: ( ((f /^ n) /. i) `1 = ((f /^ n) /. (i + 1)) `1 or ((f /^ n) /. i) `2 = ((f /^ n) /. (i + 1)) `2 )
A5: len (f /^ n) = (len f) - n by A2, RFINSEQ:def 2;
i in dom (f /^ n) by A3, A4, GOBOARD2:3;
then A6: (f /^ n) /. i = f /. (n + i) by FINSEQ_5:30;
i + 1 in dom (f /^ n) by A3, A4, GOBOARD2:3;
then A7: (f /^ n) /. (i + 1) = f /. (n + (i + 1)) by FINSEQ_5:30
.= f /. ((n + i) + 1) ;
i <= n + i by NAT_1:11;
then A8: 1 <= n + i by A3, XXREAL_0:2;
n + (i + 1) <= len f by A4, A5, XREAL_1:21;
hence ( ((f /^ n) /. i) `1 = ((f /^ n) /. (i + 1)) `1 or ((f /^ n) /. i) `2 = ((f /^ n) /. (i + 1)) `2 ) by A1, A6, A7, A8, TOPREAL1:def 7; :: thesis: verum

end;

suppose n > len f ; :: thesis: f /^ n is special

then f /^ n = <*> the carrier of (TOP-REAL 2) by RFINSEQ:def 2;
hence f /^ n is special ; :: thesis: verum

end;

end;