let f be FinSequence of (); :: thesis: for p being Point of ()
for k being Nat st f is unfolded & k + 1 = len f & (LSeg (f,k)) /\ (LSeg ((f /. (len f)),p)) = {(f /. (len f))} holds
f ^ <*p*> is unfolded

let p be Point of (); :: thesis: for k being Nat st f is unfolded & k + 1 = len f & (LSeg (f,k)) /\ (LSeg ((f /. (len f)),p)) = {(f /. (len f))} holds
f ^ <*p*> is unfolded

let k be Nat; :: thesis: ( f is unfolded & k + 1 = len f & (LSeg (f,k)) /\ (LSeg ((f /. (len f)),p)) = {(f /. (len f))} implies f ^ <*p*> is unfolded )
set g = <*p*>;
assume that
A1: f is unfolded and
A2: k + 1 = len f and
A3: (LSeg (f,k)) /\ (LSeg ((f /. (len f)),p)) = {(f /. (len f))} ; :: thesis: f ^ <*p*> is unfolded
A4: len <*p*> = 1 by FINSEQ_1:39;
A5: <*p*> /. 1 = p by FINSEQ_4:16;
A6: len (f ^ <*p*>) = (len f) + () by FINSEQ_1:22;
let i be Nat; :: according to TOPREAL1:def 6 :: thesis: ( not 1 <= i or not i + 2 <= len (f ^ <*p*>) or (LSeg ((f ^ <*p*>),i)) /\ (LSeg ((f ^ <*p*>),(i + 1))) = {((f ^ <*p*>) /. (i + 1))} )
assume that
A7: 1 <= i and
A8: i + 2 <= len (f ^ <*p*>) ; :: thesis: (LSeg ((f ^ <*p*>),i)) /\ (LSeg ((f ^ <*p*>),(i + 1))) = {((f ^ <*p*>) /. (i + 1))}
A9: i + (1 + 1) = (i + 1) + 1 ;
per cases ( i + 2 <= len f or i + 2 > len f ) ;
suppose A10: i + 2 <= len f ; :: thesis: (LSeg ((f ^ <*p*>),i)) /\ (LSeg ((f ^ <*p*>),(i + 1))) = {((f ^ <*p*>) /. (i + 1))}
then A11: i + 1 in dom f by ;
i + 1 <= (i + 1) + 1 by NAT_1:11;
hence (LSeg ((f ^ <*p*>),i)) /\ (LSeg ((f ^ <*p*>),(i + 1))) = (LSeg (f,i)) /\ (LSeg ((f ^ <*p*>),(i + 1))) by
.= (LSeg (f,i)) /\ (LSeg (f,(i + 1))) by A9, A10, Th6
.= {(f /. (i + 1))} by A1, A7, A10
.= {((f ^ <*p*>) /. (i + 1))} by ;
:: thesis: verum
end;
suppose i + 2 > len f ; :: thesis: (LSeg ((f ^ <*p*>),i)) /\ (LSeg ((f ^ <*p*>),(i + 1))) = {((f ^ <*p*>) /. (i + 1))}
then A12: len f <= i + 1 by ;
A13: not f is empty by ;
then A14: len f in dom f by FINSEQ_5:6;
i + 1 <= len f by ;
then A15: i + 1 = len f by ;
then A16: LSeg ((f ^ <*p*>),(i + 1)) = LSeg ((f /. (len f)),(<*p*> /. 1)) by ;
LSeg ((f ^ <*p*>),i) = LSeg (f,k) by A2, A15, Th6;
hence (LSeg ((f ^ <*p*>),i)) /\ (LSeg ((f ^ <*p*>),(i + 1))) = {((f ^ <*p*>) /. (i + 1))} by ; :: thesis: verum
end;
end;