reserve A,B,C for Ordinal;
reserve a,b,c,d for natural Ordinal;
reserve l,m,n for natural Ordinal;
reserve i,j,k for Element of omega;
reserve x,y,z for Element of RAT+;
reserve i,j,k for natural Ordinal;
reserve r,s,t for Element of RAT+;

theorem Th72:
  for i being ordinal Element of RAT+ st i < x & x < i+one holds not x in omega
proof
  let i be ordinal Element of RAT+;
  assume that
A1: i < x and
A2: x < i+one and
A3: x in omega;
  consider z such that
A4: i+one = x+z by A2,Def13;
  i+one = i+^1 by Th58;
  then i+one in omega by Th31;
  then reconsider z9 = z as Element of omega by A3,A4,Th71;
  consider y such that
A5: x = i+y by A1,Def13;
  i in omega by Th31;
  then reconsider y9 = y as Element of omega by A3,A5,Th71;
  i+one = i+(y+z) by A5,A4,Th51;
  then 1 = y+z by Th62
    .= y9+^z9 by Th58;
  then y9 c= 1 by ORDINAL3:24;
  then y = {} or y = 1 by ORDINAL3:16;
  then i = x or i+one = x by A5,Th50;
  hence contradiction by A1,A2;
end;
