reserve P,P1,P2,R for Subset of TOP-REAL 2,
  p,p1,p2,p3,p11,p22,q,q1,q2,q3,q4 for Point of TOP-REAL 2,
  f,h for FinSequence of TOP-REAL 2,
  r for Real,
  u for Point of Euclid 2,
  n,m,i,j,k for Nat,
  x,y for set;

theorem Th10:
  p <> q & p in Ball(u,r) & q in Ball(u,r) implies ex P st P
  is_S-P_arc_joining p,q & P c= Ball(u,r)
proof
  assume that
A1: p<>q and
A2: p in Ball(u,r) & q in Ball(u,r);
  now
    per cases by A1,TOPREAL3:6;
    suppose
A3:   p`1 <> q`1;
      now
        per cases;
        suppose
A4:       p`2 = q`2;
          reconsider P = L~<* p,|[(p`1+q`1)/2,p`2]|,q *> as Subset of TOP-REAL
          2;
          take P;
          thus P is_S-P_arc_joining p,q & P c= Ball(u,r) by A2,A3,A4,Th7;
        end;
        suppose
A5:       p`2 <> q`2;
A6:       p = |[p`1,p`2]| & q=|[q`1,q`2]| by EUCLID:53;
          now
            per cases by A2,A6,TOPREAL3:25;
            suppose
A7:           |[p`1,q`2]| in Ball(u,r);
              reconsider P = L~<* p,|[p`1,q`2]|,q *> as Subset of TOP-REAL 2;
              take P;
              thus P is_S-P_arc_joining p,q & P c= Ball(u,r) by A2,A3,A5,A7,Th8
;
            end;
            suppose
A8:           |[q`1,p`2]| in Ball(u,r);
              reconsider P = L~<* p,|[q`1,p`2]|,q *> as Subset of TOP-REAL 2;
              take P;
              thus P is_S-P_arc_joining p,q & P c= Ball(u,r) by A2,A3,A5,A8,Th9
;
            end;
          end;
          hence thesis;
        end;
      end;
      hence thesis;
    end;
    suppose
A9:   p`2 <> q`2;
      now
        per cases;
        suppose
A10:      p`1 = q`1;
          reconsider P = L~<* p,|[p`1,(p`2+q`2)/2]|,q *> as Subset of TOP-REAL
          2;
          take P;
          thus P is_S-P_arc_joining p,q & P c= Ball(u,r) by A2,A9,A10,Th6;
        end;
        suppose
A11:      p`1 <> q`1;
A12:      p = |[p`1,p`2]| & q=|[q`1,q`2]| by EUCLID:53;
          now
            per cases by A2,A12,TOPREAL3:25;
            suppose
A13:          |[p`1,q`2]| in Ball(u,r);
              reconsider P = L~<* p,|[p`1,q`2]|,q *> as Subset of TOP-REAL 2;
              take P;
              thus P is_S-P_arc_joining p,q & P c= Ball(u,r) by A2,A9,A11,A13
,Th8;
            end;
            suppose
A14:          |[q`1,p`2]| in Ball(u,r);
              reconsider P = L~<* p,|[q`1,p`2]|,q *> as Subset of TOP-REAL 2;
              take P;
              thus P is_S-P_arc_joining p,q & P c= Ball(u,r) by A2,A9,A11,A14
,Th9;
            end;
          end;
          hence thesis;
        end;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
