reserve C for Simple_closed_curve,
  i for Nat;
reserve R for non empty Subset of TOP-REAL 2,
  j, k, m, n for Nat;

theorem
  LMP C in Lower_Arc C & UMP C in Upper_Arc C or UMP C in Lower_Arc C &
  LMP C in Upper_Arc C
proof
A1: LMP C in South_Arc C by Th34;
A2: North_Arc C c= C by Th35;
A3: UMP C in North_Arc C by Th33;
A4: South_Arc C c= C by Th36;
  now
    per cases by A4,A1,A2,A3,JORDAN16:7;
    case
      LE LMP C, UMP C, C;
      then
      LMP C in Upper_Arc C & UMP C in Lower_Arc C or LMP C in Lower_Arc C
      & UMP C in Lower_Arc C or LMP C in Upper_Arc C & UMP C in Upper_Arc C;
      hence UMP C in Lower_Arc C & LMP C in Upper_Arc C by JORDAN21:49,50;
    end;
    case
      LE UMP C, LMP C, C;
      then
      UMP C in Upper_Arc C & LMP C in Lower_Arc C or LMP C in Lower_Arc C
      & UMP C in Lower_Arc C or LMP C in Upper_Arc C & UMP C in Upper_Arc C;
      hence LMP C in Lower_Arc C & UMP C in Upper_Arc C by JORDAN21:49,50;
    end;
  end;
  hence thesis;
end;
