reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;

theorem Th37:
  for f be Function st z in dom f holds Ext({f.z},x,y) = {Ext(f,x,y).z}
proof
  let f be Function;
  assume A1: z in dom f;
  per cases;
  suppose x in f.z;
    then Ext({f.z},x,y) = {(f.z)\/{y}} & Ext(f,x,y).z = (f.z)\/{y}
      by A1,Def5,Th29;
    hence thesis;
  end;
  suppose not x in f.z;
    then Ext({f.z},x,y) = {(f.z)} & Ext(f,x,y).z = (f.z) by A1,Def5,Th30;
    hence thesis;
  end;
end;
