 reserve o for object;
 reserve F for non almost_trivial Field;
 reserve x,a for Element of F;
reserve n for non zero Nat;

theorem Th17:
   for R being non degenerated Ring st 0 in the carrier of R
   holds R is non flat
   proof
     let R be non degenerated Ring;
A1:  the_rank_of 0 = 0 by CLASSES1:71;
     assume
A2:  0 in the carrier of R;
     per cases;
       suppose 0 = 0.R; then
         the_rank_of 1.R <> {} by CLASSES1:71;
         hence R is non flat by A1,A2;
       end;
       suppose 0 = 1.R; then
         the_rank_of 0.R <> {} by CLASSES1:71;
         hence R is non flat by A1,A2;
       end;
       suppose 0 <> 0.R & 0 <> 1.R; then
         the_rank_of 0.R <> {} by CLASSES1:71;
         hence R is non flat by A1,A2;
       end;
     end;
