let f be FinSequence of (TOP-REAL 2); :: thesis: for p being Point of (TOP-REAL 2)
for n being Nat st f is special & p in LSeg (f,n) holds
Ins (f,n,p) is special
let p be Point of (TOP-REAL 2); :: thesis: for n being Nat st f is special & p in LSeg (f,n) holds
Ins (f,n,p) is special
let n be Nat; :: thesis: ( f is special & p in LSeg (f,n) implies Ins (f,n,p) is special )
assume that
A1: f is special and
A2: p in LSeg (f,n) ; :: thesis: Ins (f,n,p) is special
A3: n + 1 <= len f by A2, TOPREAL1:def 3;
then A4: 1 <= (len f) - n by XREAL_1:19;
A5: 1 <= n by A2, TOPREAL1:def 3;
then A6: LSeg (f,n) = LSeg ((f /. n),(f /. (n + 1))) by A3, TOPREAL1:def 3;
set f1 = f | n;
set g1 = (f | n) ^ <*p*>;
set f2 = f /^ n;
set p1 = (f | n) /. (len (f | n));
set p2 = (f /^ n) /. 1;
A7: (f | n) /. (len (f | n)) = |[(((f | n) /. (len (f | n))) `1),(((f | n) /. (len (f | n))) `2)]| by EUCLID:53;
A8: n <= n + 1 by NAT_1:11;
then n <= len f by A3, XXREAL_0:2;
then 1 <= len (f /^ n) by A4, RFINSEQ:def 1;
then 1 in dom (f /^ n) by FINSEQ_3:25;
then A9: f /. (n + 1) = (f /^ n) /. 1 by FINSEQ_5:27;
A10: len (f | n) = n by A3, A8, FINSEQ_1:59, XXREAL_0:2;
then n in dom (f | n) by A5, FINSEQ_3:25;
then A11: f /. n = (f | n) /. (len (f | n)) by A10, FINSEQ_4:70;
then A12: ( ((f | n) /. (len (f | n))) `1 = ((f /^ n) /. 1) `1 or ((f | n) /. (len (f | n))) `2 = ((f /^ n) /. 1) `2 ) by A1, A5, A3, A9;
set q1 = ((f | n) ^ <*p*>) /. (len ((f | n) ^ <*p*>));
A13: (f /^ n) /. 1 = |[(((f /^ n) /. 1) `1),(((f /^ n) /. 1) `2)]| by EUCLID:53;
<*p*> /. 1 = p by FINSEQ_4:16;
then ( ((f | n) /. (len (f | n))) `1 = (<*p*> /. 1) `1 or ((f | n) /. (len (f | n))) `2 = (<*p*> /. 1) `2 ) by A2, A6, A11, A9, A12, A7, A13, TOPREAL3:11, TOPREAL3:12;
then A14: (f | n) ^ <*p*> is special by A1, Lm13;
((f | n) ^ <*p*>) /. (len ((f | n) ^ <*p*>)) = ((f | n) ^ <*p*>) /. ((len (f | n)) + 1) by FINSEQ_2:16
.= p by FINSEQ_4:67 ;
then ( (((f | n) ^ <*p*>) /. (len ((f | n) ^ <*p*>))) `1 = ((f /^ n) /. 1) `1 or (((f | n) ^ <*p*>) /. (len ((f | n) ^ <*p*>))) `2 = ((f /^ n) /. 1) `2 ) by A2, A6, A11, A9, A12, A7, A13, TOPREAL3:11, TOPREAL3:12;
then ((f | n) ^ <*p*>) ^ (f /^ n) is special by A1, A14, Lm13;
hence Ins (f,n,p) is special by FINSEQ_5:def 4; :: thesis: verum