theorem
  variables_in (H/(x,y)) c= (variables_in H \ {x}) \/ {y}
proof
  let a be object;
  assume
A1: a in variables_in (H/(x,y));
  then reconsider z = a as Variable;
  consider b being object such that
A2: b in dom (H/(x,y)) and
A3: z = H/(x,y).b by A1,FUNCT_1:def 3;
A4: dom (H/(x,y)) = dom H by Def3;
  then
A5: H.b <> x implies z = H.b by A2,A3,Def3;
  H.b = x implies z = y by A2,A3,A4,Def3;
  then z in {y} or z in rng H & not z in {0,1,2,3,4} & not z in {x} by A2,A4,A5
,Th136,FUNCT_1:def 3,TARSKI:def 1;
  then z in {y} or z in rng H \ {0,1,2,3,4} & not z in {x} by XBOOLE_0:def 5;
  then z in {y} or z in variables_in H \ {x} by XBOOLE_0:def 5;
  hence thesis by XBOOLE_0:def 3;
end;
