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 Th8:
  |.push(e,s).| = <*e*>^|.s.|
  proof
    not emp push(e,s) by Def12;
    hence |.push(e,s).| = <*top push(e,s)*>^|.pop push(e,s).| by Th6
    .= <*e*>^|.pop push(e,s).| by Def10
    .= <*e*>^|.s.| by Def11;
  end;
