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;
