
theorem Th82:
  for H being non empty lower-bounded RelStr st H is Heyting for a
  ,b being Element of H holds 'not' a >= b iff a "/\" b = Bottom H
proof
  let H be non empty lower-bounded RelStr;
  assume
A1: H is Heyting;
  let a,b be Element of H;
  hereby
    assume 'not' a >= b;
    then
A2: a "/\" b <= Bottom H by A1,Th67;
    a "/\" b >= Bottom H by A1,YELLOW_0:44;
    hence a "/\" b = Bottom H by A1,A2,ORDERS_2:2;
  end;
  assume a "/\" b = Bottom H;
  then a "/\" b <= Bottom H by A1,ORDERS_2:1;
  hence thesis by A1,Th67;
end;
