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

R_Segment (P,p1,p2,p1) = P

let p1, p2 be Point of (TOP-REAL 2); :: thesis: ( P is_an_arc_of p1,p2 implies R_Segment (P,p1,p2,p1) = P )

assume A1: P is_an_arc_of p1,p2 ; :: thesis: R_Segment (P,p1,p2,p1) = P

thus R_Segment (P,p1,p2,p1) c= P by Th20; :: according to XBOOLE_0:def 10 :: thesis: P c= R_Segment (P,p1,p2,p1)

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in P or x in R_Segment (P,p1,p2,p1) )

assume A2: x in P ; :: thesis: x in R_Segment (P,p1,p2,p1)

then reconsider p = x as Point of (TOP-REAL 2) ;

LE p1,p,P,p1,p2 by A1, A2, JORDAN5C:10;

hence x in R_Segment (P,p1,p2,p1) ; :: thesis: verum

R_Segment (P,p1,p2,p1) = P

let p1, p2 be Point of (TOP-REAL 2); :: thesis: ( P is_an_arc_of p1,p2 implies R_Segment (P,p1,p2,p1) = P )

assume A1: P is_an_arc_of p1,p2 ; :: thesis: R_Segment (P,p1,p2,p1) = P

thus R_Segment (P,p1,p2,p1) c= P by Th20; :: according to XBOOLE_0:def 10 :: thesis: P c= R_Segment (P,p1,p2,p1)

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in P or x in R_Segment (P,p1,p2,p1) )

assume A2: x in P ; :: thesis: x in R_Segment (P,p1,p2,p1)

then reconsider p = x as Point of (TOP-REAL 2) ;

LE p1,p,P,p1,p2 by A1, A2, JORDAN5C:10;

hence x in R_Segment (P,p1,p2,p1) ; :: thesis: verum