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 Th82:
  s <=' r implies s *' t <=' r *' t
proof
  given x such that
A1: r = s+x;
  take x*'t;
  thus thesis by A1,Th57;
end;
