
theorem Th12:
  for I being set, C being pcs-Str-yielding ManySortedSet of I
  for p, q being Element of pcs-union C holds p (--) q iff
  ex i being object, P being pcs-Str, p9, q9 being Element of P st
  i in I & P = C.i & p9 = p & q9 = q & p9 (--) q9
proof
  let I be set, C be pcs-Str-yielding ManySortedSet of I;
  set R = pcs-union C;
  let p, q be Element of R;
A1: dom pcs-ToleranceRels C = I by PARTFUN1:def 2;
  thus p (--) q implies
  ex i being object, P being pcs-Str, p9, q9 being Element of P st
  i in I & P = C.i & p9 = p & q9 = q & p9 (--) q9
  proof
    assume p (--) q;
    then [p,q] in the ToleranceRel of R;
    then [p,q] in Union pcs-ToleranceRels C by Def36;
    then consider Z being set such that
A2: [p,q] in Z and
A3: Z in rng pcs-ToleranceRels C by TARSKI:def 4;
    consider i being object such that
A4: i in dom pcs-ToleranceRels C and
A5: (pcs-ToleranceRels C).i = Z by A3,FUNCT_1:def 3;
    reconsider I1 = I as non empty set by A4;
    reconsider A1 = C as pcs-Str-yielding ManySortedSet of I1;
    reconsider i1 = i as Element of I1 by A4;
    reconsider P = A1.i1 as pcs-Str;
    take i, P;
    Z = the ToleranceRel of A1.i1 by A5,Def20;
    then reconsider p9 = p, q9 = q as Element of P by A2,ZFMISC_1:87;
    take p9, q9;
    thus i in I by A4;
    thus P = C.i & p9 = p & q9 = q;
    thus [p9,q9] in the ToleranceRel of P by A2,A5,Def20;
  end;
  given i being object, P being pcs-Str, p9, q9 being Element of P such that
A6: i in I and
A7: P = C.i and
A8: p9 = p and
A9: q9 = q and
A10: p9 (--) q9;
A11: [p9,q9] in the ToleranceRel of P by A10;
  reconsider I1 = I as non empty set by A6;
  reconsider i1 = i as Element of I1 by A6;
  reconsider A1 = C as pcs-Str-yielding ManySortedSet of I1;
  (pcs-ToleranceRels A1).i1 = the ToleranceRel of A1.i1 by Def20;
  then the ToleranceRel of A1.i1 in rng pcs-ToleranceRels C by A1,FUNCT_1:3;
  then [p,q] in Union pcs-ToleranceRels C by A7,A8,A9,A11,TARSKI:def 4;
  hence [p,q] in the ToleranceRel of R by Def36;
end;
