let f be FinSequence of (TOP-REAL 2); :: thesis: for p being Point of (TOP-REAL 2)
for k being Element of 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 (TOP-REAL 2); :: thesis: for k being Element of 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 Element of 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_1:56;
A5: <*p*> /. 1 = p by FINSEQ_4:25;
A6: len (f ^ <*p*>) = (len f) + (len <*p*>) by FINSEQ_1:35;
let i be Nat; :: according to TOPREAL1:def 8 :: 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 A7, GOBOARD2:4;
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 A10, Th6, XXREAL_0:2
.= (LSeg f,i) /\ (LSeg f,(i + 1)) by A9, A10, Th6
.= {(f /. (i + 1))} by A1, A7, A10, TOPREAL1:def 8
.= {((f ^ <*p*>) /. (i + 1))} by A11, FINSEQ_4:83 ;
:: 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 A9, NAT_1:13;
A13: not f is empty by A4, A8, A6, A9, XREAL_1:8;
then A14: len f in dom f by FINSEQ_5:6;
i + 1 <= len f by A4, A8, A6, A9, XREAL_1:8;
then A15: i + 1 = len f by A12, XXREAL_0:1;
then A16: LSeg (f ^ <*p*>),(i + 1) = LSeg (f /. (len f)),(<*p*> /. 1) by A13, Th8;
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 A3, A5, A15, A16, A14, FINSEQ_4:83; :: thesis: verum

end;

end;