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;

theorem Th54:
  x <> {} implies ex y st x*'y = 1
proof
  set nx = numerator x, dx = denominator x;
A1: dx <> {} by Th35;
  assume x <> {};
  then
A2: nx <> {} by Th37;
  take y = dx/nx;
  nx, dx are_coprime by Th34;
  then denominator y = nx & numerator y = dx by A2,Th43;
  hence thesis by A2,A1,Th41,ORDINAL3:31;
end;
