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 Th37:
  for S being stack of X/== holds emp S iff S = the s_empty of X
  proof let S be stack of X/==;
    the carrier' of X/== = Class(==_X)by Def20; then
    S in Class(==_X); then
    consider x being object such that
A1: x in the carrier' of X & S = Class(==_X,x) by EQREL_1:def 3;
    reconsider x as stack of X by A1;
    hereby assume
      emp S; then
      emp x by A1,Th36;
      hence S = the s_empty of X by A1,Th19;
    end;
    assume S = the s_empty of X; then
    x in the s_empty of X by A1,EQREL_1:20;then
    emp x;
    hence thesis by A1,Th36;
  end;
