
theorem Th19:
  for P, Q being pcs holds P misses Q implies pcs-sum(P,Q) is pcs-compatible
proof
  let P, Q be pcs;
  set D1 = the carrier of P;
  set D2 = the carrier of Q;
  set R1 = the InternalRel of P;
  set R2 = the InternalRel of Q;
  set T1 = the ToleranceRel of P;
  set T2 = the ToleranceRel of Q;
  set R = R1 \/ R2;
  set T = T1 \/ T2;
  assume
A1: D1 misses D2;
  let p, p9, q, q9 be Element of pcs-sum(P,Q) such that
A2: p (--) q and
A3: p9 <= p and
A4: q9 <= q;
A5: pcs-sum(P,Q) = pcsSUM(P,Q) by Th15;
  then
A6: [p,q] in T by A2;
  per cases by A6,XBOOLE_0:def 3;
  suppose
A7: [p,q] in T1;
    then
A8: p in D1 by ZFMISC_1:87;
A9: q in D1 by A7,ZFMISC_1:87;
    reconsider p1 = p, q1 = q as Element of P by A7,ZFMISC_1:87;
A10: p1 (--) q1 by A7;
A11: [p9,p] in R by A3,A5;
A12: [q9,q] in R by A4,A5;
    then reconsider p91 = p9, q91 = q9 as Element of P by A1,A8,A9,A11,Lm1;
A13: [p9,p] in R1 by A1,A8,A11,Lm1;
A14: [q9,q] in R1 by A1,A9,A12,Lm1;
A15: p91 <= p1 by A13;
    q91 <= q1 by A14;
    then p91 (--) q91 by A10,A15,Def22;
    then [p91,q91] in T1;
    then [p91,q91] in T by XBOOLE_0:def 3;
    hence p9 (--) q9 by A5;
  end;
  suppose
A16: [p,q] in T2;
    then
A17: p in D2 by ZFMISC_1:87;
A18: q in D2 by A16,ZFMISC_1:87;
    reconsider p1 = p, q1 = q as Element of Q by A16,ZFMISC_1:87;
A19: p1 (--) q1 by A16;
A20: [p9,p] in R by A3,A5;
A21: [q9,q] in R by A4,A5;
    then reconsider p91 = p9, q91 = q9 as Element of Q by A1,A17,A18,A20,Lm1;
A22: [p9,p] in R2 by A1,A17,A20,Lm1;
A23: [q9,q] in R2 by A1,A18,A21,Lm1;
A24: p91 <= p1 by A22;
    q91 <= q1 by A23;
    then p91 (--) q91 by A19,A24,Def22;
    then [p91,q91] in T2;
    then [p91,q91] in T by XBOOLE_0:def 3;
    hence p9 (--) q9 by A5;
  end;
end;
