reserve r,s,t,x9,y9,z9,p,q for Element of RAT+;
reserve x,y,z for Element of REAL+;

theorem
  { A where A is Subset of RAT+: r in A implies (for s st s <=' r holds
  s in A) & ex s st s in A & r < s} is c=-linear
proof
  set IR = { A where A is Subset of RAT+: r in A implies (for s st s <=' r
  holds s in A) & ex s st s in A & r < s};
  let x,y be set;
  assume x in IR;
  then
A1: ex A9 being Subset of RAT+ st x = A9 & for r holds r in A9 implies (for s
  st s <=' r holds s in A9) & ex s st s in A9 & r < s;
  assume y in IR;
  then
A2: ex B9 being Subset of RAT+ st y = B9 & for r holds r in B9 implies (for s
  st s <=' r holds s in B9) & ex s st s in B9 & r < s;
  assume not x c= y;
  then consider s being object such that
A3: s in x and
A4: not s in y;
  reconsider s as Element of RAT+ by A1,A3;
  let e be object;
  assume
A5: e in y;
  then reconsider r = e as Element of RAT+ by A2;
  r <=' s by A2,A4,A5;
  hence thesis by A1,A3;
end;
