
theorem Th1:
  for R being non empty Poset, A,B being ManySortedSet of the
  carrier of R, OR being ManySortedRelation of A,B holds OR is os-compatible
  implies OR is OrderSortedSet of R
proof
  let R be non empty Poset, A,B be ManySortedSet of the carrier of R, OR be
  ManySortedRelation of A,B;
  set OR1 = OR;
  assume
A1: OR is os-compatible;
  OR1 is order-sorted
  proof
    let s1,s2 be Element of R such that
A2: s1 <= s2;
    for x,y being object holds [x,y] in OR.s1 implies [x,y] in OR.s2
    proof
      let x,y be object such that
A3:   [x,y] in OR.s1;
      x in A.s1 & y in B.s1 by A3,ZFMISC_1:87;
      hence thesis by A1,A2,A3;
    end;
    hence thesis by RELAT_1:def 3;
  end;
  hence thesis;
end;
