reserve i,j for Nat;
reserve x,y for set;
reserve A for non empty set;
reserve c for Element of StandardStackSystem A;
reserve m for stack of StandardStackSystem A;
reserve X for non empty non void StackSystem;
reserve s,s1,s2 for stack of X;
reserve e,e1,e2 for Element of X;

theorem Th2:
  for X being non empty non void StackSystem st X is pop-finite
  ex s being stack of X st emp s
  proof
    let X be non empty non void StackSystem such that
A1: X is pop-finite;
    set s1 = the stack of X;
    defpred P[set, stack of X, stack of X] means $3 = pop $2;
A2: for n being Nat for x being stack of X
    ex y being stack of X st P[n,x,y];
    consider f being sequence of the carrier' of X such that
A3: f.0 = s1 & for n being Nat holds P[n,f.n,f.(n+1)]
    from RECDEF_1:sch 2(A2);
    consider i being Nat, s being stack of X such that
A4: f.i = s & (not emp s implies f.(i+1) <> pop s) by A1;
    take s;
    thus thesis by A3,A4;
  end;
