reserve a, b for Int_position,
  i for Instruction of SCMPDS,
  l for Element of NAT,
  k, k1, k2 for Integer;

theorem Th17:
  JUMP (return a) = {k where k is Nat: k > 1}
proof
  set A = {k where k is Nat: k > 1};
  set i = return a;
  set X = the set of all  NIC(i,l) where l is Nat;
  JUMP i c= NIC(i,(0)) by AMISTD_1:19;
  hence JUMP i c= A by Th4;
  let x be object;
  assume
A1: x in A;
  now
    consider k being Nat such that
A2: x = k and
A3: k > 1 by A1;
    reconsider k2 = k-2 as Element of NAT by A3,Lm2;
    NIC(i,(0)) in X;
    hence X <> {};
    a in SCM-Data-Loc by AMI_2:def 16;
    then consider j being Nat such that
A4: a = [1,j] by AMI_2:23;
    set t = [1,j+1];
    set s = the State of SCMPDS;
    let y be set;
A5: DataLoc(j,1) = [1,|.j+1.|]
      .= t by ABSVALUE:def 1;
    t in SCM-Data-Loc by AMI_2:24;
    then reconsider t1 = t as Int_position by AMI_2:def 16;
    assume y in X;
    then consider l being Nat such that
A6: y = NIC(i,l);
  reconsider n = l as Element of NAT by ORDINAL1:def 12;
    set I = i;
  reconsider u = the n-started State of SCMPDS
     as Element of product the_Values_of SCMPDS by CARD_3:107;
A7: IC u = n by MEMSTR_0:def 11;
    set g = (a,t1)-->(j,k2);
  reconsider v = u +* g
     as Element of product the_Values_of SCMPDS by CARD_3:107;
    j <> j+1;
    then
A8: a <> t1 by A4,XTUPLE_0:1;
    then
A9: v.a = j by FUNCT_4:84;
A10: v.t1 = k2 by A8,FUNCT_4:84;
A11: dom g = {a,t1} by FUNCT_4:62;
    a <> IC SCMPDS & t1 <> IC SCMPDS by SCMPDS_2:43;
    then
A12: not IC SCMPDS in dom g by A11,TARSKI:def 2;
A13: IC v = l by A7,A12,FUNCT_4:11;
    x = ((k2 ) + 2) by A2
      .= (|.v.DataLoc(j,1).| ) + 2 by A10,A5,ABSVALUE:def 1
      .= IC Exec(i,v) by A9,SCMPDS_2:58,SCMPDS_I:def 14;
    hence x in y by A6,A13;
  end;
  hence thesis by SETFAM_1:def 1;
end;
