let t1 be Real; :: thesis: for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : t1 < t } holds
P is convex
let P be Subset of (TOP-REAL 2); :: thesis: ( P = { |[s,t]| where s, t is Real : t1 < t } implies P is convex )
assume A1: P = { |[s,t]| where s, t is Real : t1 < t } ; :: thesis: P is convex
let w1 be Point of (TOP-REAL 2); :: according to JORDAN1:def 1 :: thesis: for w2 being Element of (TOP-REAL 2) st w1 in P & w2 in P holds
LSeg (w1,w2) c= P
let w2 be Point of (TOP-REAL 2); :: thesis: ( w1 in P & w2 in P implies LSeg (w1,w2) c= P )
assume that
A2: w1 in P and
A3: w2 in P ; :: thesis: LSeg (w1,w2) c= P
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in LSeg (w1,w2) or x in P )
assume A4: x in LSeg (w1,w2) ; :: thesis: x in P
consider s3, t3 being Real such that
A5: |[s3,t3]| = w1 and
A6: t1 < t3 by A1, A2;
A7: w1 `2 = t3 by A5, EUCLID:52;
consider s4, t4 being Real such that
A8: |[s4,t4]| = w2 and
A9: t1 < t4 by A1, A3;
A10: w2 `2 = t4 by A8, EUCLID:52;
consider l being Real such that
A11: x = ((1 - l) * w1) + (l * w2) and
A12: 0 <= l and
A13: l <= 1 by A4;
set w = ((1 - l) * w1) + (l * w2);
A14: ((1 - l) * w1) + (l * w2) = |[((((1 - l) * w1) `1) + ((l * w2) `1)),((((1 - l) * w1) `2) + ((l * w2) `2))]| by EUCLID:55;
A15: (1 - l) * w1 = |[((1 - l) * (w1 `1)),((1 - l) * (w1 `2))]| by EUCLID:57;
A16: l * w2 = |[(l * (w2 `1)),(l * (w2 `2))]| by EUCLID:57;
A17: ((1 - l) * w1) `2 = (1 - l) * (w1 `2) by A15, EUCLID:52;
(l * w2) `2 = l * (w2 `2) by A16, EUCLID:52;
then (((1 - l) * w1) + (l * w2)) `2 = ((1 - l) * (w1 `2)) + (l * (w2 `2)) by A14, A17, EUCLID:52;
then A18: t1 < (((1 - l) * w1) + (l * w2)) `2 by A6, A7, A9, A10, A12, A13, XREAL_1:175;
((1 - l) * w1) + (l * w2) = |[((((1 - l) * w1) + (l * w2)) `1),((((1 - l) * w1) + (l * w2)) `2)]| by EUCLID:53;
hence x in P by A1, A11, A18; :: thesis: verum