theorem Th75:
  x in C & p in east_halfline x /\ L~Cage(C,n) implies p`1 > x`1
proof
  set f = Cage(C,n);
  assume
A1: x in C;
  assume
A2: p in east_halfline x /\ L~f;
  then
A3: p in east_halfline x by XBOOLE_0:def 4;
  then
A4: p`2 = x`2 by TOPREAL1:def 11;
  assume
A5: p`1 <= x`1;
  p`1 >= x`1 by A3,TOPREAL1:def 11;
  then p`1 = x`1 by A5,XXREAL_0:1;
  then
A6: p = x by A4,TOPREAL3:6;
  p in L~f by A2,XBOOLE_0:def 4;
  then x in C /\ L~f by A1,A6,XBOOLE_0:def 4;
  then C meets L~f;
  hence contradiction by JORDAN10:5;
end;
