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 Th79:
  r <=' s *' t implies ex t0 being Element of RAT+ st r = s *' t0 & t0 <=' t
proof
  given x such that
A1: s*'t = r+x;
  per cases;
  suppose
A2: s = {};
    take t;
    s*'t = {} by A2,Th48;
    hence thesis by A1,Th63;
  end;
  suppose
A3: s <> {};
    then consider t1 being Element of RAT+ such that
A4: x = s*'t1 by Th55;
    consider t0 being Element of RAT+ such that
A5: r = s*'t0 by A3,Th55;
    take t0;
    thus r = s*'t0 by A5;
    take t1;
    s*'t = s*'(t0+t1) by A1,A5,A4,Th57;
    hence thesis by A3,Th56;
  end;
end;
