reserve x for Real;

theorem Th6:
  for a,b,c,x be Real st x in ].a,c.[ holds x in ].a,b.[ or
  x = b or x in ].b,c.[
proof
  let a,b,c,x be Real;
  assume that
A1: x in ].a,c.[ and
A2: not x in ].a,b.[ and
A3: not x = b;
  x <= a or x >= b by A2,XXREAL_1:4;
  then
A4: x > b by A1,A3,XXREAL_0:1,XXREAL_1:4;
  x < c by A1,XXREAL_1:4;
  hence thesis by A4,XXREAL_1:4;
end;
