let s, t2 be Real; for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where t is Real : t < t2 } holds
P is convex
let P be Subset of (TOP-REAL 2); ( P = { |[s,t]| where t is Real : t < t2 } implies P is convex )
assume A1:
P = { |[s,t]| where t is Real : t < t2 }
; P is convex
let w1, w2 be Point of (TOP-REAL 2); JORDAN1:def 1 ( not w1 in P or not w2 in P or LSeg (w1,w2) c= P )
assume that
A2:
w1 in P
and
A3:
w2 in P
; LSeg (w1,w2) c= P
consider t3 being Real such that
A4:
|[s,t3]| = w1
and
A5:
t3 < t2
by A1, A2;
consider t4 being Real such that
A6:
|[s,t4]| = w2
and
A7:
t4 < t2
by A1, A3;
A8:
w2 `2 = t4
by A6, EUCLID:52;
let x be object ; TARSKI:def 3 ( not x in LSeg (w1,w2) or x in P )
assume
x in LSeg (w1,w2)
; x in P
then consider l being Real such that
A9:
x = ((1 - l) * w1) + (l * w2)
and
A10:
( 0 <= l & l <= 1 )
;
set w = ((1 - l) * w1) + (l * w2);
A11:
((1 - l) * w1) + (l * w2) = |[((((1 - l) * w1) `1) + ((l * w2) `1)),((((1 - l) * w1) `2) + ((l * w2) `2))]|
by EUCLID:55;
A12:
l * w2 = |[(l * (w2 `1)),(l * (w2 `2))]|
by EUCLID:57;
then A13:
(l * w2) `1 = l * (w2 `1)
by EUCLID:52;
A14:
(1 - l) * w1 = |[((1 - l) * (w1 `1)),((1 - l) * (w1 `2))]|
by EUCLID:57;
then
((1 - l) * w1) `1 = (1 - l) * (w1 `1)
by EUCLID:52;
then A15:
(((1 - l) * w1) + (l * w2)) `1 = ((1 - l) * (w1 `1)) + (l * (w2 `1))
by A11, A13, EUCLID:52;
w2 `1 = s
by A6, EUCLID:52;
then A16: (((1 - l) * w1) + (l * w2)) `1 =
((1 - l) * s) + (l * s)
by A4, A15, EUCLID:52
.=
s - 0
;
A17:
((1 - l) * w1) + (l * w2) = |[((((1 - l) * w1) + (l * w2)) `1),((((1 - l) * w1) + (l * w2)) `2)]|
by EUCLID:53;
A18:
(l * w2) `2 = l * (w2 `2)
by A12, EUCLID:52;
((1 - l) * w1) `2 = (1 - l) * (w1 `2)
by A14, EUCLID:52;
then A19:
(((1 - l) * w1) + (l * w2)) `2 = ((1 - l) * (w1 `2)) + (l * (w2 `2))
by A11, A18, EUCLID:52;
w1 `2 = t3
by A4, EUCLID:52;
then
t2 > (((1 - l) * w1) + (l * w2)) `2
by A5, A7, A8, A10, A19, XREAL_1:176;
hence
x in P
by A1, A9, A17, A16; verum