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 Th11:
  for s being stack of StandardStackSystem A holds |.s.| = s
  proof
    defpred P[stack of StandardStackSystem A] means |.$1.| = $1;
A1: now let s be stack of StandardStackSystem A;
      assume emp s; then
      s = {} & |.s.| = {} by Def7,Th5;
      hence P[s];
    end;
A2: now let s be stack of StandardStackSystem A;
      let e be Element of StandardStackSystem A;
      assume P[s]; then
      |.push(e,s).| = <*e*>^s by Th8;
      hence P[push(e,s)] by Def7;
    end;
    let s be stack of StandardStackSystem A;
    thus P[s] from INDsch(A1,A2);
  end;
