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

theorem
  for P be non empty pcs-Str st
    the ToleranceRel of P = id the carrier of P holds
      P is pcs-Compatible
  proof
    let P be non empty pcs-Str;
    assume
A1: the ToleranceRel of P = id the carrier of P;
    for a1,a2,b1,b2 being Element of P holds
      a1 (--) b1 & a2 (--) b2
    implies
      (a1 "\/" a2) (--) (b1 "\/" b2) & (a1 "/\" a2) (--) (b1 "/\" b2)
    proof
      let a1,a2,b1,b2 be Element of P;
      assume a1 (--) b1 & a2 (--) b2; then
      [a1,b1] in the ToleranceRel of P &
        [a2,b2] in the ToleranceRel of P by PCS_0:def 7; then
      a1 = b1 & a2 = b2 by A1,RELAT_1:def 10; then
      [a1 "\/" a2,b1 "\/" b2] in id the carrier of P &
        [a1 "/\" a2,b1 "/\" b2] in id the carrier of P
          by RELAT_1:def 10;
      hence thesis by A1,PCS_0:def 7;
    end;
    hence thesis;
  end;
