
theorem Th50:
  for L being complete antisymmetric non empty RelStr
  for F be Function-yielding Function holds
    \//(F, L) = /\\(F, L opp) & /\\(F, L) = \//(F, L opp)
proof
  let L be complete antisymmetric non empty RelStr;
  let F be Function-yielding Function;
A1: now
    let x be object;
    assume x in dom F;
    then \//(F,L).x = \\/(F.x,L) & /\\(F,L opp).x = //\(F.x,L opp)
      by WAYBEL_5:def 1,def 2;
    hence \//(F,L).x = /\\(F,L opp).x by Th49;
  end;
  dom \//(F,L) = dom F & dom /\\(F,L opp) = dom F by FUNCT_2:def 1;
  hence \//(F,L) = /\\(F,L opp) by A1,FUNCT_1:2;
A2: now
    let x be object;
    assume x in dom F;
    then /\\(F,L).x = //\(F.x,L) & \//(F,L opp).x = \\/(F.x,L opp)
      by WAYBEL_5:def 1,def 2;
    hence /\\(F,L).x = \//(F,L opp).x by Th49;
  end;
  dom /\\(F,L) = dom F & dom \//(F,L opp) = dom F by FUNCT_2:def 1;
  hence thesis by A2,FUNCT_1:2;
end;
