let f be FinSequence of (TOP-REAL 2); for i being Nat st f is being_S-Seq & 1 <= i & i + 1 <= len f holds
LE f /. i,f /. (i + 1), L~ f,f /. 1,f /. (len f)
let i be Nat; ( f is being_S-Seq & 1 <= i & i + 1 <= len f implies LE f /. i,f /. (i + 1), L~ f,f /. 1,f /. (len f) )
assume that
A1:
f is being_S-Seq
and
A2:
( 1 <= i & i + 1 <= len f )
; LE f /. i,f /. (i + 1), L~ f,f /. 1,f /. (len f)
set p1 = f /. 1;
set p2 = f /. (len f);
set q1 = f /. i;
set q2 = f /. (i + 1);
A3:
len f >= 2
by A1, TOPREAL1:def 8;
then reconsider P = L~ f as non empty Subset of (TOP-REAL 2) by TOPREAL1:23;
i + 1 in dom f
by A2, SEQ_4:134;
then A4:
f /. (i + 1) in P
by A3, GOBOARD1:1;
A5:
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 = f /. (i + 1) & 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 = f /. (i + 1) & 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 = f /. (i + 1) & 0 <= s2 & s2 <= 1 implies s1 <= s2 )
assume that A6:
g is
being_homeomorphism
and A7:
(
g . 0 = f /. 1 &
g . 1
= f /. (len f) )
and A8:
g . s1 = f /. i
and A9:
(
0 <= s1 &
s1 <= 1 )
and A10:
g . s2 = f /. (i + 1)
and A11:
(
0 <= s2 &
s2 <= 1 )
;
s1 <= s2
A12:
dom g =
[#] I[01]
by A6, TOPS_2:def 5
.=
the
carrier of
I[01]
;
then A13:
s1 in dom g
by A9, BORSUK_1:43;
A14:
s2 in dom g
by A11, A12, BORSUK_1:43;
A15:
g is
one-to-one
by A6, TOPS_2:def 5;
consider r1,
r2 being
Real such that A16:
r1 < r2
and A17:
0 <= r1
and A18:
r1 <= 1
and
0 <= r2
and A19:
r2 <= 1
and
LSeg (
f,
i)
= g .: [.r1,r2.]
and A20:
g . r1 = f /. i
and A21:
g . r2 = f /. (i + 1)
by A1, A2, A6, A7, JORDAN5B:7;
A22:
r2 in dom g
by A16, A17, A19, A12, BORSUK_1:43;
r1 in dom g
by A17, A18, A12, BORSUK_1:43;
then
s1 = r1
by A8, A20, A13, A15, FUNCT_1:def 4;
hence
s1 <= s2
by A10, A16, A21, A22, A14, A15, FUNCT_1:def 4;
verum
end;
i in dom f
by A2, SEQ_4:134;
then
f /. i in P
by A3, GOBOARD1:1;
hence
LE f /. i,f /. (i + 1), L~ f,f /. 1,f /. (len f)
by A4, A5; verum