let P be Subset of (TOP-REAL 2); 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); ( P is_an_arc_of p1,p2 implies R_Segment (P,p1,p2,p2) = {p2} )
assume A1:
P is_an_arc_of p1,p2
; R_Segment (P,p1,p2,p2) = {p2}
then A2:
p2 in P
by TOPREAL1:1;
thus
R_Segment (P,p1,p2,p2) c= {p2}
XBOOLE_0:def 10 {p2} c= R_Segment (P,p1,p2,p2)proof
let x be
object ;
TARSKI:def 3 ( not x in R_Segment (P,p1,p2,p2) or x in {p2} )
assume
x in R_Segment (
P,
p1,
p2,
p2)
;
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;
verum
end;
let x be object ; TARSKI:def 3 ( not x in {p2} or x in R_Segment (P,p1,p2,p2) )
assume
x in {p2}
; 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; verum