
theorem Th7:
  for R being RelStr, x,y being Element of R
  st R is quasi_ordered holds x in Class(EqRel R, y) iff x <= y & y <= x
proof
  let R be RelStr, x,y be Element of R such that
A1: R is quasi_ordered;
  set IR = the InternalRel of R;
  hereby
    assume x in Class(EqRel R,y);
    then [x,y] in EqRel R by EQREL_1:19;
    then
A2: [x,y] in IR /\ IR~ by A1,Def4;
    then
A3: [x,y] in IR by XBOOLE_0:def 4;
A4: [x,y] in IR~ by A2,XBOOLE_0:def 4;
    thus x <= y by A3;
    [y,x] in IR by A4,RELAT_1:def 7;
    hence y <= x;
  end;
  assume that
A5: x <= y and
A6: y <= x;
A7: [y,x] in IR by A6;
A8: [x,y] in IR by A5;
  [x,y] in IR~ by A7,RELAT_1:def 7;
  then [x,y] in IR /\ IR~ by A8,XBOOLE_0:def 4;
  then [x,y] in EqRel R by A1,Def4;
  hence thesis by EQREL_1:19;
end;
