let t2 be Real; :: thesis: for P being Subset of (TOP-REAL 2) st P = { |[s,t]| where s, t is Real : t < t2 } holds
P is convex
let P be Subset of (TOP-REAL 2); :: thesis: ( P = { |[s,t]| where s, t is Real : t < t2 } implies P is convex )
assume A1: P = { |[s,t]| where s, t is Real : t < t2 } ; :: 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 set ; :: 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: t3 < t2 by A1, A2;
A7: w1 `2 = t3 by A5, EUCLID:56;
consider s4, t4 being Real such that
A8: |[s4,t4]| = w2 and
A9: t4 < t2 by A1, A3;
A10: w2 `2 = t4 by A8, EUCLID:56;
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:59;
A15: (1 - l) * w1 = |[((1 - l) * (w1 `1 )),((1 - l) * (w1 `2 ))]| by EUCLID:61;
A16: l * w2 = |[(l * (w2 `1 )),(l * (w2 `2 ))]| by EUCLID:61;
A17: ((1 - l) * w1) `2 = (1 - l) * (w1 `2 ) by A15, EUCLID:56;
(l * w2) `2 = l * (w2 `2 ) by A16, EUCLID:56;
then (((1 - l) * w1) + (l * w2)) `2 = ((1 - l) * (w1 `2 )) + (l * (w2 `2 )) by A14, A17, EUCLID:56;
then A18: t2 > (((1 - l) * w1) + (l * w2)) `2 by A6, A7, A9, A10, A12, A13, XREAL_1:178;
((1 - l) * w1) + (l * w2) = |[((((1 - l) * w1) + (l * w2)) `1 ),((((1 - l) * w1) + (l * w2)) `2 )]| by EUCLID:57;
hence x in P by A1, A11, A18; :: thesis: verum