reserve x,y for set;

theorem
  for F being Field, a being Element of NonZero F,
      b,c being Element of F holds a*b = a*c implies b = c
proof
  let F be Field, a be Element of NonZero F, b,c be Element of suppf1(F);
  reconsider x=a as Element of F;
  assume
A1: a*b = a*c;
  reconsider ra = revf(F).a as Element of F by XBOOLE_0:def 5;
  b = 1.F*b
    .= omf(F).(1.F,b)
    .= (x*ra)*b by Def6
    .= ra*(x*c) by A1,Th19
    .= (x*ra)*c by Th19
    .= omf(F).(1.F,c) by Def6
    .= 1.F*c
    .= c;
  hence thesis;
end;
