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 Th33:
  ex s st (coset s1)/\Class(==_X, s2) = {s}
  proof
    consider s such that
A1: |.s.| = |.s2.| & s in coset s1 by Th30;
    take s;
    thus (coset s1)/\Class(==_X, s2) c= {s}
    proof let x be object; assume
A2:   x in (coset s1)/\Class(==_X, s2); then
A3:   x in coset s1 & x in Class(==_X, s2) by XBOOLE_0:def 4;
      reconsider x as stack of X by A2;
      [s2,x] in ==_X by A3,EQREL_1:18; then
      s2 == x by Def16; then
      |.s2.| = |.x.|; then
      s = x by A1,A3,Th32;
      hence thesis by TARSKI:def 1;
    end;
    s == s2 by A1; then
    [s2,s] in ==_X by Def16; then
    s in Class(==_X, s2) by EQREL_1:18; then
    {s} c= Class(==_X, s2) & {s} c= coset s1 by A1,ZFMISC_1:31;
    hence thesis by XBOOLE_1:19;
  end;
