let f be FinSequence of (); :: thesis: for q being Point of ()
for i being Nat st f is being_S-Seq & 1 <= i & i + 1 <= len f & q in LSeg (f,i) holds
LE f /. i,q, L~ f,f /. 1,f /. (len f)

let q be Point of (); :: thesis: for i being Nat st f is being_S-Seq & 1 <= i & i + 1 <= len f & q in LSeg (f,i) holds
LE f /. i,q, L~ f,f /. 1,f /. (len f)

let i be Nat; :: thesis: ( f is being_S-Seq & 1 <= i & i + 1 <= len f & q in LSeg (f,i) implies LE f /. i,q, L~ f,f /. 1,f /. (len f) )
assume that
A1: f is being_S-Seq and
A2: ( 1 <= i & i + 1 <= len f ) and
A3: q in LSeg (f,i) ; :: thesis: LE f /. i,q, L~ f,f /. 1,f /. (len f)
set p1 = f /. 1;
set p2 = f /. (len f);
set q1 = f /. i;
A4: 2 <= len f by ;
then reconsider P = L~ f as non empty Subset of () by TOPREAL1:23;
i in dom f by ;
then A5: f /. i in P by ;
A6: for g being Function of I,(() | P)
for s1, s2 being Real st g is being_homeomorphism & g . 0 = f /. 1 & g . 1 = f /. (len f) & g . s1 = f /. i & 0 <= s1 & s1 <= 1 & g . s2 = q & 0 <= s2 & s2 <= 1 holds
s1 <= s2
proof
let g be Function of I,(() | P); :: thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = f /. 1 & g . 1 = f /. (len f) & g . s1 = f /. i & 0 <= s1 & s1 <= 1 & g . s2 = q & 0 <= s2 & s2 <= 1 holds
s1 <= s2

let s1, s2 be Real; :: thesis: ( g is being_homeomorphism & g . 0 = f /. 1 & g . 1 = f /. (len f) & g . s1 = f /. i & 0 <= s1 & s1 <= 1 & g . s2 = q & 0 <= s2 & s2 <= 1 implies s1 <= s2 )
assume that
A7: g is being_homeomorphism and
A8: ( g . 0 = f /. 1 & g . 1 = f /. (len f) ) and
A9: g . s1 = f /. i and
A10: ( 0 <= s1 & s1 <= 1 ) and
A11: g . s2 = q and
A12: ( 0 <= s2 & s2 <= 1 ) ; :: thesis: s1 <= s2
A13: dom g = [#] I by
.= the carrier of I ;
then A14: s1 in dom g by ;
consider r1, r2 being Real such that
r1 < r2 and
A15: ( 0 <= r1 & r1 <= 1 ) and
0 <= r2 and
r2 <= 1 and
A16: LSeg (f,i) = g .: [.r1,r2.] and
A17: g . r1 = f /. i and
g . r2 = f /. (i + 1) by ;
consider r39 being object such that
A18: r39 in dom g and
A19: r39 in [.r1,r2.] and
A20: g . r39 = q by ;
r39 in { l where l is Real : ( r1 <= l & l <= r2 ) } by ;
then consider r3 being Real such that
A21: r3 = r39 and
A22: r1 <= r3 and
r3 <= r2 ;
A23: g is one-to-one by ;
A24: r1 in dom g by ;
s2 in dom g by ;
then s2 = r3 by ;
hence s1 <= s2 by ; :: thesis: verum
end;
q in P by ;
hence LE f /. i,q, L~ f,f /. 1,f /. (len f) by A5, A6; :: thesis: verum