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,p2 being Point of TOP-REAL 2 st P is_an_arc_of p1,p2 holds
  x_Middle(P,p1,p2) in P & y_Middle(P,p1,p2) in P
proof
  let P be Subset of TOP-REAL 2, p1,p2 be Point of TOP-REAL 2;
  assume
A1: P is_an_arc_of p1,p2;
  deffunc F(Point of TOP-REAL 2)=$1;
  defpred P[Point of TOP-REAL 2] means $1`1=(p1`1+p2`1)/2;
  reconsider Q={F(q):P[q]} as Subset of TOP-REAL 2 from DOMAIN_1:sch 8;
A2: x_Middle(P,p1,p2)=First_Point(P,p1,p2,Q) by Def1;
A3: P meets Q by A1,Th13;
  P /\ Q is closed by A1,Th13;
  then
A4: x_Middle(P,p1,p2) in P /\ Q by A1,A2,A3,JORDAN5C:def 1;
  defpred Q[Point of TOP-REAL 2] means $1`2=(p1`2+p2`2)/2;
  reconsider Q2={F(q):Q[q]} as Subset of TOP-REAL 2 from DOMAIN_1:sch 8;
A5: y_Middle(P,p1,p2)=First_Point(P,p1,p2,Q2) by Def2;
A6: P meets Q2 by A1,Th14;
  P /\ Q2 is closed by A1,Th14;
  then y_Middle(P,p1,p2) in P /\ Q2 by A1,A5,A6,JORDAN5C:def 1;
  hence thesis by A4,XBOOLE_0:def 4;
end;
