reserve X,Y for set, x,y,z for object, i,j,n for natural number;
reserve
  n for non empty Nat,
  S for non empty non void n PC-correct PCLangSignature,
  L for language MSAlgebra over S,
  F for PC-theory of L,
  A,B,C,D for Formula of L;
reserve
  J for non empty non void Signature,
  T for non-empty MSAlgebra over J,
  X for non empty-yielding GeneratorSet of T,
  S1 for J-extension non empty non void n PC-correct QC-correct
  QCLangSignature over Union X,
  L for non-empty Language of X extended_by ({},the carrier of S1), S1,
  G for QC-theory of L,
  A,B,C,D for Formula of L;
reserve x,y,z for Element of Union X;
reserve x0,y0,z0 for Element of Union (X extended_by ({},the carrier of S1));
reserve a for SortSymbol of J;
reserve
  L for
    non-empty T-extension Language of X extended_by ({},the carrier of S1), S1,
  G for QC-theory of L,
  G1 for QC-theory_with_equality of L,
  A,B,C,D for Formula of L,
  s,s1 for SortSymbol of S1,
  t,t9 for Element of L,s,
  t1,t2,t3 for Element of L,s1;
reserve
  n for non empty natural number,
  J for non empty non void Signature,
  T for non-empty VarMSAlgebra over J,
  X for non-empty GeneratorSet of T,
  S for essential J-extension non empty non void n PC-correct QC-correct
  n AL-correct AlgLangSignature over Union X,
  L for non empty IfWhileAlgebra of X,S,
  M,M1,M2 for Algorithm of L,
  A,B,C,V for Formula of L,
  H for AL-theory of V,L,
  a for SortSymbol of J,
  x,y for (Element of X.a),
  t for Element of T,a;

theorem
  \Cap(M,A) \iff A\and(M*A)\and\Cap(M, (M\;M)*A) in H
  proof
A1: \Cap(M,A) \iff A\and\Cap(M, M*A) in H by Def43;
    \Cap(M,M*A) \iff (M*A)\and\Cap(M, M*(M*A)) in H by Def43;
    then
A2: \Cap(M,A) \iff A\and((M*A)\and\Cap(M, M*(M*A))) in H by A1,Th97;
    A\and(M*A)\and\Cap(M, M*(M*A))\iffA\and((M*A)\and\Cap(M, M*(M*A))) in H
    by Th78;
    then A\and((M*A)\and\Cap(M, M*(M*A)))\iffA\and(M*A)\and\Cap(M, M*(M*A))
    in H by Th90;
    then
A3: \Cap(M,A) \iff A\and(M*A)\and\Cap(M, M*(M*A)) in H by A2,Th91;
    ((M\;M)*A)\iff(M*(M*A)) in H by Def43;
    then (M*(M*A)) \iff ((M\;M)*A) in H by Th90;
    then \Cap(M, M*(M*A))\iff\Cap(M, (M\;M)*A) in H by Th148;
    hence thesis by A3,Th97;
  end;
