 reserve W for WA-Lattice;
 reserve a,b,c for Element of W;

theorem NablaCompat:
  for P be non empty pcs-Str st
    the ToleranceRel of P = nabla the carrier of P holds
      P is pcs-Compatible
  proof
    let P be non empty pcs-Str;
    assume
A1: the ToleranceRel of P = nabla the carrier of P;
    for a1,a2,b1,b2 being Element of P holds
      (a1 "\/" a2) (--) (b1 "\/" b2) & (a1 "/\" a2) (--) (b1 "/\" b2)
    proof
      let a1,a2,b1,b2 be Element of P;
      set X = the carrier of P;
      [a1 "\/" a2,b1 "\/" b2] in [:X,X:] &
        [a1 "/\" a2,b1 "/\" b2] in [:X,X:] by ZFMISC_1:def 2; then
      [a1 "\/" a2,b1 "\/" b2] in nabla the carrier of P &
        [a1 "/\" a2,b1 "/\" b2] in nabla the carrier of P
          by EQREL_1:def 1;
      hence thesis by A1,PCS_0:def 7;
    end;
    hence thesis;
  end;
