reserve C for Simple_closed_curve,
  P for Subset of TOP-REAL 2,
  p for Point of TOP-REAL 2,
  n for Element of NAT;
reserve D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem Th19:
  W-bound C = W-bound Lower_Arc C
proof
A1: W-bound Lower_Arc C <= W-bound C
  proof
A2: (W-min C)`1 = W-bound C by EUCLID:52;
    assume
A3: W-bound Lower_Arc C > W-bound C;
A4: west_halfline W-min C misses Lower_Arc C
    proof
      assume west_halfline W-min C meets Lower_Arc C;
      then consider p being object such that
A5:   p in west_halfline W-min C and
A6:   p in Lower_Arc C by XBOOLE_0:3;
      reconsider p as Point of TOP-REAL 2 by A5;
      p`1 >= W-bound Lower_Arc C by A6,PSCOMP_1:24;
      then W-bound C < p`1 by A3,XXREAL_0:2;
      hence contradiction by A2,A5,TOPREAL1:def 13;
    end;
    W-min C in west_halfline W-min C & W-min C in Lower_Arc C by JORDAN7:1
,TOPREAL1:38;
    hence contradiction by A4,XBOOLE_0:3;
  end;
  W-bound C <= W-bound Lower_Arc C by JORDAN6:61,PSCOMP_1:69;
  hence thesis by A1,XXREAL_0:1;
end;
