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,p2) = {p2}
let p1, p2 be Point of (TOP-REAL 2); :: thesis: ( P is_an_arc_of p1,p2 implies R_Segment (P,p1,p2,p2) = {p2} )
assume A1: P is_an_arc_of p1,p2 ; :: thesis: R_Segment (P,p1,p2,p2) = {p2}
then A2: p2 in P by TOPREAL1:1;
thus R_Segment (P,p1,p2,p2) c= {p2} :: according to XBOOLE_0:def 10 :: thesis: {p2} c= R_Segment (P,p1,p2,p2)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in R_Segment (P,p1,p2,p2) or x in {p2} )
assume x in R_Segment (P,p1,p2,p2) ; :: thesis: x in {p2}
then consider q being Point of (TOP-REAL 2) such that
A3: q = x and
A4: LE p2,q,P,p1,p2 ;
q in P by A4;
then LE q,p2,P,p1,p2 by A1, JORDAN5C:10;
then q = p2 by A1, A4, JORDAN5C:12;
hence x in {p2} by A3, TARSKI:def 1; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in {p2} or x in R_Segment (P,p1,p2,p2) )
assume x in {p2} ; :: thesis: x in R_Segment (P,p1,p2,p2)
then A5: x = p2 by TARSKI:def 1;
LE p2,p2,P,p1,p2 by A2, JORDAN5C:9;
hence x in R_Segment (P,p1,p2,p2) by A5; :: thesis: verum