reserve a,b,c,d,x,y,w,z,x1,x2,x3,x4 , X for set;
reserve A for non empty set;
reserve i,j,k for Element of NAT;
reserve a,b,c,d for Real;
reserve y,r,s,x,t,w for Element of RAT+;

theorem Th3:
  for A being Subset of RAT+ st (ex t st t in A & t <> {}) &
  for r,s st r in A & s <=' r holds s in A
  ex r1,r2,r3,r4,r5 being Element of RAT+ st
  r1 in A & r2 in A & r3 in A & r4 in A & r5 in A &
  r1 <> r2 & r1 <> r3 & r1<>r4 & r1 <> r5 & r2<> r3 & r2<> r4 &
  r2<> r5& r3 <> r4 & r3 <> r5 & r4 <> r5
proof
  let A be Subset of RAT+;
  given t such that
A1: t in A and
A2: t <> {};
  assume
A3: for r,s st r in A & s <=' r holds s in A;
  consider x such that
A4: t = x+x by ARYTM_3:60;
  consider y such that
A5: x = y+y by ARYTM_3:60;
  consider w such that
A6: y = w+w by ARYTM_3:60;
  take {},w,y,x,t;
A7: {} <=' t by ARYTM_3:64;
A8: x <=' t by A4;
A9: y <=' x by A5;
A10: w<=' y by A6;
A11: y <=' t by A8,A9,ARYTM_3:67;
  w <=' x by A9,A10,ARYTM_3:67;
  hence {} in A & w in A & y in A & x in A & t in A by A1,A3,A7,A8,A9,
ARYTM_3:67;
A12: {} <> x by A2,A4,ARYTM_3:50;
  then
A13: {} <> y by A5,ARYTM_3:50;
A14: x <> t
  proof
    assume x = t;
    then t+{} = t+t by A4,ARYTM_3:50;
    hence contradiction by A2,ARYTM_3:62;
  end;
A15: y <> x
  proof
    assume y = x;
    then x+{} = x+x by A5,ARYTM_3:50;
    hence contradiction by A12,ARYTM_3:62;
  end;
  w <> y
  proof
    assume w = y;
    then y+{} = y+y by A6,ARYTM_3:50;
    hence contradiction by A13,ARYTM_3:62;
  end;
  hence thesis by A2,A4,A5,A6,A8,A9,A10,A11,A14,A15,ARYTM_3:50,66;
end;
