reserve C for Simple_closed_curve,
  p,q,p1 for Point of TOP-REAL 2,
  i,j,k,n for Nat,
  r,s for Real;

theorem Th15:
  L_Segment(Lower_Arc C,E-max C,W-min C,p) =
  Segment(Lower_Arc C,E-max C,W-min C,E-max C,p)
proof
  Lower_Arc C is_an_arc_of E-max C,W-min C by JORDAN6:50;
  then R_Segment(Lower_Arc C,E-max C,W-min C,E-max C) = Lower_Arc C
  by JORDAN6:24;
  hence Segment(Lower_Arc C,E-max C,W-min C,E-max C,p) =
  Lower_Arc C /\ L_Segment(Lower_Arc C,E-max C,W-min C,p) by JORDAN6:def 5
    .= L_Segment(Lower_Arc C,E-max C,W-min C,p) by JORDAN6:19,XBOOLE_1:28;
end;
