reserve J,J1,K for Element of Segm 13,
  b,b1,b2,c,c1,c2 for Element of SCM+FSA-Data-Loc,
  f,f1,f2 for Element of SCM+FSA-Data*-Loc;
reserve k for Nat,
  J,K,L for Element of Segm 13,
  O,P,R for Element of Segm 9;
reserve da for Int-Location,
  fa for FinSeq-Location,
  x,y for set;
reserve la,lb for Nat,
  La for Nat,
  i for Instruction of SCM+FSA,
  I for Instruction of SCM,
  l for Nat,
  LA,LB for Nat,
  dA,dB,dC,dD for Element of SCM+FSA-Data-Loc,
  DA,DB,DC for Element of SCM-Data-Loc,
  fA,fB,fC for Element of SCM+FSA-Data*-Loc,
  f,g for FinSeq-Location,
  A,B for Data-Location,
  a,b,c,db for Int-Location;
reserve S for State of SCM,
  s,s1 for State of SCM+FSA;

theorem Th65:
  Exec(c:=(g,a), s).IC SCM+FSA = IC s + 1 & (ex k st k = |.s.a.|
& Exec(c:=(g,a), s).c = (s.g)/.k) & (for b st b <> c holds Exec(c:=(g,a), s).b
  = s.b) & for f holds Exec(c:=(g,a), s).f = s.f
proof
  reconsider p = g as Element of SCM+FSA-Data*-Loc by Def3;
  reconsider mk = a, ml = c as Element of SCM-Data-Loc by AMI_2:def 16;
  reconsider I = c:=(g,a) as Element of SCM+FSA-Instr;
  reconsider S = s as SCM+FSA-State by CARD_3:107;
  reconsider J = 9 as Element of Segm 13 by NAT_1:44;
  InsCode I = 9;
  then consider i being Integer, k being Nat such that
A1: k = |.S.(I int_addr2).| and
A2: i = (S.(I coll_addr1))/.k and
A3: SCM+FSA-Exec-Res(I,S) = SCM+FSA-Chg(SCM+FSA-Chg(S,I int_addr1,i),
  IC S + 1) by SCMFSA_1:def 16;
  set S1 = SCM+FSA-Chg(S,I int_addr1,i);
A4: Exec(c:=(g,a), s) = (SCM+FSA-Chg(S1, IC S + 1)) by A3,SCMFSA_1:def 17;
  hence Exec(c:=(g,a), s).IC SCM+FSA = IC s + 1 by Th1,SCMFSA_1:19;
A5: I = [J,{},<*ml,p,mk*>] & I`3_3 = <*ml,p,mk*>;
  then
A6: I int_addr1 = ml by SCMFSA_I:def 3;
A7: I coll_addr1 = p by A5,SCMFSA_I:def 5;
  hereby
     reconsider k as Nat;
    take k;
    thus k = |.s.a.| by A5,A1,SCMFSA_I:def 4;
    thus Exec(c:=(g,a), s).c = S1.ml by A4,SCMFSA_1:20
      .= (s.g)/.k by A2,A6,A7,SCMFSA_1:24;
  end;
  hereby
    let b;
    reconsider mn = b as Element of SCM-Data-Loc by AMI_2:def 16;
    assume
A8: b <> c;
    thus Exec(c:=(g,a), s).b = S1.mn by A4,SCMFSA_1:20
      .= s.b by A6,A8,SCMFSA_1:25;
  end;
  let f;
  reconsider q = f as Element of SCM+FSA-Data*-Loc by Def3;
  thus Exec(c:=(g,a), s).f = S1.q by A4,SCMFSA_1:21
    .= s.f by SCMFSA_1:26;
end;
