reserve            x for object,
               X,Y,Z for set,
         i,j,k,l,m,n for Nat,
                 r,s for Real,
                  no for Element of OrderedNAT,
                   A for Subset of [:NAT,NAT:];

theorem Th5:
  0 < r implies ex m st m is non zero & 1 / m < r
  proof
    assume
A1: 0 < r;
    consider m be Nat such that
A2: 1/r < m by SEQ_4:3;
A3: 0 < m by A1,A2;
    take m;
    thus m is non zero by A1,A2;
    1/r*r < m*r by A2,A1,XREAL_1:68;
    then 1 < m*r by A1,XCMPLX_1:106;
    then 1/m < m*r/m by A3,XREAL_1:74;
    then 1/m < r*(m/m) by XCMPLX_1:74;
    hence thesis by A3,XCMPLX_1:88;
  end;
