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
  for x0,y0 being Element of Union (X extended_by ({},the carrier of S))
  st x = x0 & y = y0 holds
  ((x:=(@y,L))*A) \iff (A/(x0,y0)) in H
  proof
    let x0,y0 be Element of Union (X extended_by ({},the carrier of S));
    reconsider b = a as SortSymbol of S by Th8;
    reconsider t = @y as Element of (the Sorts of L).b by Th16;
    assume A1: x = x0 & y = y0;
    then
A2: ((x:=(@y,L))*A) \iff (A/(x0,t)) in H by Def43;
A3: X extended_by ({}, the carrier of S) is ManySortedSubset of the Sorts of L
    by Th23;
    a in the carrier of J = dom X by PARTFUN1:def 2;
    then b in dom(X|the carrier of S) by RELAT_1:57;
    then (X extended_by ({}, the carrier of S)).b = (X|the carrier of S).b
    by FUNCT_4:13 .= X.b by FUNCT_1:49;
    hence thesis by A1,A2,A3,Th14;
  end;
