reserve X,Z for set;
reserve x,y,z for object;
reserve A,B,C for Subset of X;

theorem Th93:
  for A being Preorder, B being Subset of A st
      the InternalRel of A is_connected_in B
    holds
      the InternalRel of QuotientOrder(A) is_connected_in (proj A).:B
proof
  let A be Preorder, B be Subset of A;
  set qa = QuotientOrder(A);
  set car = the carrier of A;
  set carq = the carrier of qa;
  set int = the InternalRel of A;
  set intq = the InternalRel of qa;
  set C = (proj A).:B;
  assume A1: int is_connected_in B;
  for X, Y being object holds
    X in C & Y in C & X <> Y implies [X,Y] in intq or [Y,X] in intq
  proof
    let X, Y be object;
    assume that
      A2: X in C & Y in C and
      A3: X <> Y;
    consider x being object such that
      A4: x in dom proj A and
      A5: x in B and
      A6: X = (proj A).x by A2, FUNCT_1:def 6;
    consider y being object such that
      A7: y in dom proj A and
      A8: y in B and
      A9: Y = (proj A).y by A2, FUNCT_1:def 6;
    per cases;
    suppose A is empty;
      hence thesis by A2;
    end;
    suppose A is non empty;
      then reconsider A as non empty Preorder;
      reconsider x, y as Element of A by A4, A7;
      x <> y by A3, A6, A9;
      then [x,y] in int or [y,x] in int by A1, A5, A8, RELAT_2:def 6;
      then (proj A).x <= (proj A).y or (proj A).y <= (proj A).x by Th45,
        ORDERS_2:def 5;
      hence [X,Y] in intq or [Y,X] in intq by A6, A9, ORDERS_2:def 5;
    end;
  end;
  hence thesis by RELAT_2:def 6;
end;
