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;
reserve X1,X2,X3 for StackAlgebra;
reserve F,F1,F2,G,G1,G2 for Function;

theorem
  for X being proper-for-identity StackAlgebra holds
  X, StandardStackSystem the carrier of X are_isomorphic
  proof
    let X be proper-for-identity StackAlgebra;
    consider G such that
    (for s being stack of X holds G.s = |.s.|) and
A1: id the carrier of X, G form_isomorphism_between
    X, StandardStackSystem the carrier of X by Th48;
    take id the carrier of X, G;
    thus thesis by A1;
  end;
