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;
reserve C for Simple_closed_curve;

theorem Th62:
  Lower_Arc C meets Vertical_Line((W-bound C+E-bound C)/2)
proof
A1: W-bound C <= E-bound C by SPRECT_1:21;
  (W-min C)`1 = W-bound C by EUCLID:52;
  then
A2: (W-min C)`1<=(W-bound C +E-bound C)/2 by A1,Th1;
  (E-max C)`1 = E-bound C by EUCLID:52;
  then
A3: (W-bound C +E-bound C)/2<=(E-max C)`1 by A1,Th1;
  Lower_Arc C is_an_arc_of W-min C,E-max C by Th50;
  hence thesis by A2,A3,Th49;
end;
