
theorem Th30:
  for P being transitive pcs-Str, a being set st not a in the carrier of P
  holds pcs-extension(P,a) is transitive
proof
  let P be transitive pcs-Str, a be set such that
A1: not a in the carrier of P;
  set R = pcs-extension(P,a);
A2: the InternalRel of R = [:{a},the carrier of R:] \/ the InternalRel of P
  by Def39;
  let x, y, z be object;
  assume that
A3: x in the carrier of R and
A4: y in the carrier of R and
A5: z in the carrier of R and
A6: [x,y] in the InternalRel of R and
A7: [y,z] in the InternalRel of R;
A8: [a,z] in [:{a},the carrier of R:] by A5,ZFMISC_1:105;
  reconsider x, y, z as Element of R by A3,A4,A5;
A9: x <= y by A6;
A10: y <= z by A7;
  per cases;
  suppose x = a;
    hence thesis by A2,A8,XBOOLE_0:def 3;
  end;
  suppose
A11: x <> a;
    then reconsider x0 = x as Element of P by Th25;
A12: x0 <> a by A11;
    then reconsider y0 = y as Element of P by A1,A9,Th24;
    y0 <> a by A1,A9,A12,Th24;
    then reconsider z0 = z as Element of P by A1,A10,Th24;
A13: y <> a by A1,A9,A12,Th24;
A14: x0 <= y0 by A9,A11,Th26;
    y0 <= z0 by A10,A13,Th26;
    then x0 <= z0 by A14,YELLOW_0:def 2;
    then x <= z by Th23;
    hence thesis;
  end;
end;
