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
  for P being Subset of TOP-REAL 2, P1 being Subset of (TOP-REAL 2)|P
  st P is being_simple_closed_curve
  & Upper_Arc(P) /\ P1={W-min(P),E-max(P)} & Upper_Arc(P) \/ P1=P
  holds P1=Lower_Arc(P)
proof
  let P be Subset of TOP-REAL 2, P1 be Subset of (TOP-REAL 2)|P;
  assume that
A1: P is being_simple_closed_curve and
A2: Upper_Arc(P) /\ P1={W-min(P),E-max(P)} and
A3: Upper_Arc(P) \/ P1=P;
  set B=Upper_Arc(P);
  (B \/ P1 \B)\/ (B /\ P1) =(P1 \ B) \/ (P1/\B) by XBOOLE_1:40
    .=P1 by XBOOLE_1:51;
  hence thesis by A1,A2,A3,Th51;
end;
