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

theorem
  x = y + z implies z <=' x
proof
  reconsider zz = {} as Element of REAL+ by Th1;
  assume
A1: x = y + z;
  assume
A2: not z <=' x;
  then consider y0 being Element of REAL+ such that
A3: x + y0 = z by Th9;
  x = x + (y + y0) by A1,A3,Th6;
  then x + zz = x + (y + y0) by Def8;
  then y0 = {} by Th5,Th11;
  then z = x by A3,Def8;
  hence thesis by A2;
end;
