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 Th99:
 for x,y being object holds rng(F+*(x,y)) c= rng F \/ {y}
proof let x,y be object;
A1: rng(x.-->y) = {y} by FUNCOP_1:8;
  per cases;
  suppose
    x in dom F;
    then F+*(x,y) = F+*(x.-->y) by Def2;
    hence thesis by A1,FUNCT_4:17;
  end;
  suppose
    not x in dom F;
    then F+*(x,y) = F by Def2;
    hence thesis by XBOOLE_1:7;
  end;
end;
