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, TOPREAL3:44; :: 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:57;
then |[(((p `1 ) + (q `1 )) / 2),(p `2 )]| in Ball u,r by A2, A3, TOPREAL3:31;
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, TOPREAL3:28;
thus L~ f is_S-P_arc_joining p,q by A5, Def1; :: 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, TOPREAL3:23;
hence L~ f c= Ball u,r by A6, XBOOLE_1:8; :: thesis: verum