reserve x,y for set;
reserve s,r for Real;
reserve r1,r2 for Real;
reserve n for Nat;
reserve p,q,q1,q2 for Point of TOP-REAL 2;

theorem Th28:
  for P being Subset of TOP-REAL 2, p1,p2,q being Point of TOP-REAL 2 st
  P is_an_arc_of p1,p2
  holds L_Segment(P,p1,p2,q)=R_Segment(P,p2,p1,q)
proof
  let P be Subset of TOP-REAL 2, p1,p2,q be Point of TOP-REAL 2;
  assume that
A1: P is_an_arc_of p1,p2;
  thus L_Segment(P,p1,p2,q) c= R_Segment(P,p2,p1,q)
  proof
    let x be object;
    assume x in L_Segment(P,p1,p2,q);
    then consider p such that
A2: p=x and
A3: LE p,q,P,p1,p2;
    LE q,p,P,p2,p1 by A1,A3,Th18;
    hence thesis by A2;
  end;
  let x be object;
  assume x in R_Segment(P,p2,p1,q);
  then consider p such that
A4: p=x and
A5: LE q,p,P,p2,p1;
  LE p,q,P,p1,p2 by A1,A5,Th18,JORDAN5B:14;
  hence thesis by A4;
end;
