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

theorem
  x <> {} implies ex y st x *' y = one
proof
  reconsider rone =one as Element of REAL+ by Th1;
  assume x <> {};
  then DEDEKIND_CUT x <> {} by Lm10;
  then consider B being Element of DEDEKIND_CUTS such that
A1: DEDEKIND_CUT x *' B = DEDEKIND_CUT rone by Lm43;
  take y = GLUED B;
  thus x *' y = GLUED DEDEKIND_CUT rone by A1,Lm12
    .= one by Lm23;
end;
