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;
reserve X for StackAlgebra;
reserve s,s1,s2,s3 for stack of X;
reserve e,e1,e2,e3 for Element of X;

theorem Th19:
  emp s implies Class(==_X, s) = the s_empty of X
  proof
    assume
A1: emp s;
    thus Class(==_X, s) c= the s_empty of X
    proof
      let x be object; assume
A2:   x in Class(==_X, s); then
      reconsider s1 = x as stack of X;
      [s,s1] in ==_X by A2,EQREL_1:18; then
      s == s1 by Def16; then
      emp s1 by A1,Th14;
      hence thesis;
    end;
    let x be object; assume
A3: x in the s_empty of X; then
    reconsider s1 = x as stack of X;
    emp s1 by A3; then
    s == s1 by A1,Th15; then
    [s,s1] in ==_X by Def16;
    hence thesis by EQREL_1:18;
  end;
