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 Th54:
  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 &
  LE q,p1,P,p1,p2 holds q=p1
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 and
A2: LE q,p1,P,p1,p2;
  q in P by A2;
  then LE p1,q,P,p1,p2 by A1,JORDAN5C:10;
  hence thesis by A1,A2,JORDAN5C:12;
end;
