reserve X,Y for set, x,y,z for object, i,j,n for natural number;

theorem Th24:
  for J being non void Signature
  for T being MSAlgebra over J
  for X being ManySortedSubset of the Sorts of T
  for S being J-extension non void Signature holds
  Union (X extended_by ({},the carrier of S)) = Union X
  proof
    let J be non void Signature;
    let T be MSAlgebra over J;
    let X be ManySortedSubset of the Sorts of T;
    let S be J-extension non void Signature;
    set Y = X extended_by ({},the carrier of S);
A1: J is Subsignature of S by Def2;
    dom X = the carrier of J by PARTFUN1:def 2;
    then
A2: X|the carrier of S = X by A1,RELAT_1:68,INSTALG1:10;
    then rng Y c= rng ((the carrier of S)-->{}) \/ rng X by FUNCT_4:17;
    then Union Y c= union({{}} \/ rng X) = union{{}} \/ Union X = {} \/ Union X
    by ZFMISC_1:77,78;
    hence Union Y c= Union X;
    X c= Y by A2,FUNCT_4:25;
    hence Union X c= Union Y by RELAT_1:11,ZFMISC_1:77;
  end;
