let P be Subset of (TOP-REAL 2); :: thesis: for p1, p2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 holds
L_Segment P,p1,p2,p2 = P
let p1, p2 be Point of (TOP-REAL 2); :: thesis: ( P is_an_arc_of p1,p2 implies L_Segment P,p1,p2,p2 = P )
assume A1: P is_an_arc_of p1,p2 ; :: thesis: L_Segment P,p1,p2,p2 = P
thus L_Segment P,p1,p2,p2 c= P by Th20; :: according to XBOOLE_0:def 10 :: thesis: P c= L_Segment P,p1,p2,p2
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in P or x in L_Segment P,p1,p2,p2 )
assume A2: x in P ; :: thesis: x in L_Segment P,p1,p2,p2
then reconsider p = x as Point of (TOP-REAL 2) ;
LE p,p2,P,p1,p2 by A1, A2, JORDAN5C:10;
hence x in L_Segment P,p1,p2,p2 ; :: thesis: verum