reserve a,x,y for object, A,B for set,
  l,m,n for Nat;
reserve X,Y for set, x for object,
  p,q for Function-yielding FinSequence,
  f,g,h for Function;
reserve m,n,k for Nat, R for Relation;
reserve i,j for Nat;
reserve F for Function,
  e,x,y,z for object;

theorem Th100:
 for x,y being object holds rng F c= rng(F+*(x,y)) \/ {F.x}
proof let x,y be object;
  let z be object;
  assume z in rng F;
  then consider e being object such that
A1: e in dom F and
A2: z = F.e by FUNCT_1:def 3;
A3: dom F = dom(F+*(x,y)) by Th29;
  per cases;
  suppose
    e = x;
    then z in {F.x} by A2,TARSKI:def 1;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose
    e <> x;
    then (F+*(x,y)).e = F.e by Th31;
    then z in rng(F+*(x,y)) by A1,A2,A3,FUNCT_1:3;
    hence thesis by XBOOLE_0:def 3;
  end;
end;
