theorem Th229:
  a <> 0 & A///a c= B///a implies A c= B
proof
  assume that
A1: a <> 0 and
A2: A///a c= B///a;
  let z;
  assume z in A;
  then z/a in A///a by Th226;
  then ex c st z/a = c/a & c in B by A2,Th228;
  hence thesis by A1,XCMPLX_1:5;
end;
