let f be FinSequence of (TOP-REAL 2); for q being Point of (TOP-REAL 2)
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 (TOP-REAL 2); 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; ( 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)
; 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 A1, TOPREAL1:def 8;
then reconsider P = L~ f as non empty Subset of (TOP-REAL 2) by TOPREAL1:23;
i in dom f
by A2, SEQ_4:134;
then A5:
f /. i in P
by A4, GOBOARD1:1;
A6:
for g being Function of I[01],((TOP-REAL 2) | 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[01],
((TOP-REAL 2) | 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 <= s2let s1,
s2 be
Real;
( 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 )
;
s1 <= s2
A13:
dom g =
[#] I[01]
by A7, TOPS_2:def 5
.=
the
carrier of
I[01]
;
then A14:
s1 in dom g
by A10, BORSUK_1:43;
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 A1, A2, A7, A8, JORDAN5B:7;
consider r39 being
object such that A18:
r39 in dom g
and A19:
r39 in [.r1,r2.]
and A20:
g . r39 = q
by A3, A16, FUNCT_1:def 6;
r39 in { l where l is Real : ( r1 <= l & l <= r2 ) }
by A19, RCOMP_1:def 1;
then consider r3 being
Real such that A21:
r3 = r39
and A22:
r1 <= r3
and
r3 <= r2
;
A23:
g is
one-to-one
by A7, TOPS_2:def 5;
A24:
r1 in dom g
by A15, A13, BORSUK_1:43;
s2 in dom g
by A12, A13, BORSUK_1:43;
then
s2 = r3
by A11, A18, A20, A21, A23, FUNCT_1:def 4;
hence
s1 <= s2
by A9, A17, A22, A24, A14, A23, FUNCT_1:def 4;
verum
end;
q in P
by A3, SPPOL_2:17;
hence
LE f /. i,q, L~ f,f /. 1,f /. (len f)
by A5, A6; verum