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 Th51:
  for P being Subset of TOP-REAL 2 st P is being_simple_closed_curve
  holds Lower_Arc(P)=(P\Upper_Arc(P)) \/ {W-min(P),E-max(P)} &
  Upper_Arc(P)=(P\Lower_Arc(P)) \/ {W-min(P),E-max(P)}
proof
  let P be Subset of TOP-REAL 2;
  assume
A1: P is being_simple_closed_curve;
  then
A2: Upper_Arc(P) /\ Lower_Arc(P)={W-min(P),E-max(P)} by Def9;
  set B=Upper_Arc(P),A=Lower_Arc(P);
A3: (B \/ A \B)\/ (B /\ A) =(A \ B) \/ (A/\B) by XBOOLE_1:40
    .=A by XBOOLE_1:51;
  (B \/ A \A)\/ (B /\ A) =(B \ A) \/ (B/\A) by XBOOLE_1:40
    .=B by XBOOLE_1:51;
  hence thesis by A1,A2,A3,Def9;
end;
