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 Th56:
  x <> {} & x*'y = x*'z implies y = z
proof
  assume x <> {};
  then consider r being Element of RAT+ such that
A1: x*'r = 1 by Th54;
  r*'(x*'y) = one*'y by A1,Th52;
  then
A2: r*'(x*'y) = y by Th53;
  r*'(x*'z) = one*'z by A1,Th52;
  hence thesis by A2,Th53;
end;
