reserve p,p1,p2,p3,q,q1,q2 for Point of TOP-REAL 2,
  i for Nat,
  lambda for Real;

theorem Th47:
  for a,b,c,d being Real st a<b & c <d
  holds LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
  is_an_arc_of W-min rectangle(a,b,c,d), E-max rectangle(a,b,c,d) &
  LSeg(|[a,c]|,|[b,c]|) \/ LSeg(|[b,c]|,|[b,d]|)
  is_an_arc_of E-max rectangle(a,b,c,d), W-min rectangle(a,b,c,d)
proof
  let a,b,c,d be Real;
  set K = rectangle(a,b,c,d);
  assume that
A1: a<b and
A2: c <d;
A3: W-min(K)= |[a,c]| by A1,A2,Th46;
A4: E-max(K)= |[b,d]| by A1,A2,Th46;
  (|[a,c]|)`2=c by EUCLID:52;
  then
A5: |[a,c]| <> |[a,d]| by A2,EUCLID:52;
  set p1= |[a,c]|,p2= |[a,d]|,q1=|[b,d]|;
A6: LSeg(p1,p2) /\ LSeg(p2,q1) ={p2} by A1,A2,Th34;
  (|[a,c]|)`1=a by EUCLID:52;
  then
A7: |[a,c]| <> |[b,c]| by A1,EUCLID:52;
  set q2=|[b,c]|;
  LSeg(q1,q2) /\ LSeg(q2,p1) ={q2} by A1,A2,Th32;
  hence thesis by A3,A4,A5,A6,A7,TOPREAL1:12;
end;
