
theorem Th8:
  for L being RelStr, x being Element of L, X being set holds (x
  is_<=_than X iff x~ is_>=_than X) & (x is_>=_than X iff x~ is_<=_than X)
proof
  let L be RelStr, x be Element of L, X be set;
A1: now
    let L be RelStr, x be Element of L, X be set;
    assume
A2: x is_<=_than X;
    thus x~ is_>=_than X
    proof
      let a be Element of L opp;
      assume a in X;
      then (~a)~ = ~a & x <= ~a by A2;
      hence thesis by LATTICE3:9;
    end;
  end;
A3: now
    let L be RelStr, x be Element of L, X be set;
    assume
A4: x is_>=_than X;
    thus x~ is_<=_than X
    proof
      let a be Element of L opp;
      assume a in X;
      then (~a)~ = ~a & x >= ~a by A4;
      hence thesis by LATTICE3:9;
    end;
  end;
  x~~ is_<=_than X implies x is_<=_than X by YELLOW_0:2;
  hence x is_<=_than X iff x~ is_>=_than X by A1,A3;
  x~~ is_>=_than X implies x is_>=_than X by YELLOW_0:2;
  hence thesis by A1,A3;
end;
