let p, q be Point of (TOP-REAL 2); :: thesis: for f being FinSequence of (TOP-REAL 2)
for r being Real
for u being Point of (Euclid 2) st p `1 <> q `1 & p `2 = q `2 & p in Ball (u,r) & q in Ball (u,r) & f = <*p,|[(((p `1) + (q `1)) / 2),(p `2)]|,q*> holds
( f is being_S-Seq & f /. 1 = p & f /. (len f) = q & L~ f is_S-P_arc_joining p,q & L~ f c= Ball (u,r) )
let f be FinSequence of (TOP-REAL 2); :: thesis: for r being Real
for u being Point of (Euclid 2) st p `1 <> q `1 & p `2 = q `2 & p in Ball (u,r) & q in Ball (u,r) & f = <*p,|[(((p `1) + (q `1)) / 2),(p `2)]|,q*> holds
( f is being_S-Seq & f /. 1 = p & f /. (len f) = q & L~ f is_S-P_arc_joining p,q & L~ f c= Ball (u,r) )
let r be Real; :: thesis: for u being Point of (Euclid 2) st p `1 <> q `1 & p `2 = q `2 & p in Ball (u,r) & q in Ball (u,r) & f = <*p,|[(((p `1) + (q `1)) / 2),(p `2)]|,q*> holds
( f is being_S-Seq & f /. 1 = p & f /. (len f) = q & L~ f is_S-P_arc_joining p,q & L~ f c= Ball (u,r) )
let u be Point of (Euclid 2); :: thesis: ( p `1 <> q `1 & p `2 = q `2 & p in Ball (u,r) & q in Ball (u,r) & f = <*p,|[(((p `1) + (q `1)) / 2),(p `2)]|,q*> implies ( f is being_S-Seq & f /. 1 = p & f /. (len f) = q & L~ f is_S-P_arc_joining p,q & L~ f c= Ball (u,r) ) )
assume that
A1: p `1 <> q `1 and
A2: p `2 = q `2 and
A3: ( p in Ball (u,r) & q in Ball (u,r) ) and
A4: f = <*p,|[(((p `1) + (q `1)) / 2),(p `2)]|,q*> ; :: thesis: ( f is being_S-Seq & f /. 1 = p & f /. (len f) = q & L~ f is_S-P_arc_joining p,q & L~ f c= Ball (u,r) )
thus A5: ( f is being_S-Seq & f /. 1 = p & f /. (len f) = q ) by A1, A2, A4, TOPREAL3:37; :: thesis: ( L~ f is_S-P_arc_joining p,q & L~ f c= Ball (u,r) )
( p = |[(p `1),(p `2)]| & q = |[(q `1),(q `2)]| ) by EUCLID:53;
then |[(((p `1) + (q `1)) / 2),(p `2)]| in Ball (u,r) by A2, A3, TOPREAL3:24;
then A6: ( LSeg (p,|[(((p `1) + (q `1)) / 2),(p `2)]|) c= Ball (u,r) & LSeg (|[(((p `1) + (q `1)) / 2),(p `2)]|,q) c= Ball (u,r) ) by A3, TOPREAL3:21;
thus L~ f is_S-P_arc_joining p,q by A5; :: thesis: L~ f c= Ball (u,r)
L~ f = (LSeg (p,|[(((p `1) + (q `1)) / 2),(p `2)]|)) \/ (LSeg (|[(((p `1) + (q `1)) / 2),(p `2)]|,q)) by A4, TOPREAL3:16;
hence L~ f c= Ball (u,r) by A6, XBOOLE_1:8; :: thesis: verum