reserve a,a1,a2,v,v1,v2,x for object;
reserve V,A for set;
reserve m,n for Nat;
reserve S,S1,S2 for FinSequence;
reserve D,D1,D2 for NonatomicND of V,A;

theorem Th35:
  D1 tolerates D2 & S2 IsNDRankSeq V,A & S1 c= S2 &
   D1 in Union S1 & D2 in Union S2
  implies D1 \/ D2 in Union S2
  proof
    set D = D1 \/ D2;
    set S = S1+*S2;
A1: dom S = dom S1 \/ dom S2 by FUNCT_4:def 1;
    assume that
A2: D1 tolerates D2 and
A3: S2 IsNDRankSeq V,A and
A4: S1 c= S2 and
A5: D1 in Union S1 and
A6: D2 in Union S2;
    Union S1 c= Union S2 by A4,CARD_3:24;
    then consider i being object such that
A7: i in dom S2 and
A8: D1 in S2.i by A5,CARD_5:2;
    consider j being object such that
A9: j in dom S2 and
A10: D2 in S2.j by A6,CARD_5:2;
    reconsider i as Element of NAT by A7;
    reconsider j as Element of NAT by A9;
    set k = max(i,j);
    dom S1 c= dom S2 by A4,XTUPLE_0:8;
    then
A11: S = S2 by FUNCT_4:19;
    k in dom S1 or k in dom S2 by A7,A9,XXREAL_0:16;
    then
A12: k in dom S by A1,XBOOLE_0:def 3;
    then
A13: S2.i c= S2.k by A3,A11,Th25,XXREAL_0:25;
    S2.j c= S2.k by A3,A11,A12,Th25,XXREAL_0:25;
    hence thesis by A11,A12,A2,A3,A10,A8,A13,Th34,CARD_5:2;
  end;
