reserve x, y, i for object,
  L for up-complete Semilattice;

theorem Th13:
  for L being non empty RelStr for F being Function-yielding
Function holds (y in rng \//(F, L) iff ex x st x in dom F & y = \\/(F.x, L)) &
  (y in rng /\\(F, L) iff ex x st x in dom F & y = //\(F.x, L))
proof
  let L be non empty RelStr;
  let F be Function-yielding Function;
  thus y in rng \//(F, L) iff ex x st x in dom F & y = \\/(F.x, L)
  proof
    hereby
      assume y in rng \//(F, L);
      then consider x being object such that
A1:   x in dom \//(F, L) & y = \//(F, L).x by FUNCT_1:def 3;
      take x;
      thus x in dom F & y = \\/(F.x, L) by A1,Def1;
    end;
    given x such that
A2: x in dom F & y = \\/(F.x, L);
    x in dom \//(F, L) & y = \//(F, L).x by A2,Def1,FUNCT_2:def 1;
    hence thesis by FUNCT_1:def 3;
  end;
  thus y in rng /\\(F, L) iff ex x st x in dom F & y = //\(F.x, L)
  proof
    hereby
      assume y in rng /\\(F, L);
      then consider x being object such that
A3:   x in dom /\\(F, L) & y = /\\(F, L).x by FUNCT_1:def 3;
      take x;
      thus x in dom F & y = //\(F.x, L) by A3,Def2;
    end;
    given x such that
A4: x in dom F & y = //\(F.x, L);
    x in dom /\\(F, L) & y = /\\(F, L).x by A4,Def2,FUNCT_2:def 1;
    hence thesis by FUNCT_1:def 3;
  end;
end;
