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
  X is push-pop & not emp s1 & not emp s2 &
  pop s1 = pop s2 & top s1 = top s2 implies s1 = s2
  proof
    assume A1: X is push-pop;
    assume not emp s1; then
    s1 = push(top s1, pop s1) by A1;
    hence thesis by A1;
  end;
