
theorem Th7:
  for R being discrete non empty Poset, C being Component of R ex x
  being Element of R st C = {x}
proof
  let R be discrete non empty Poset, C be Component of R;
  consider x being object such that
A1: x in the carrier of R and
A2: C = Class(Path_Rel R,x) by EQREL_1:def 3;
  reconsider x1 = x as Element of R by A1;
  take x1;
  for y being object holds y in C iff y = x1
  proof
    let y be object;
    hereby
      assume
A3:   y in C;
      then reconsider y1 = y as Element of R;
      [y,x] in Path_Rel R by A2,A3,EQREL_1:19;
      then y1 = x1 by Th6;
      hence y = x1;
    end;
    assume y = x1;
    hence thesis by A2,EQREL_1:20;
  end;
  hence thesis by TARSKI:def 1;
end;
