theorem Th182:
  not x in variables_in H implies H/(x,y) = H
proof
  assume
A1: not x in variables_in H;
A2: not x in {0,1,2,3,4} by Th136;
A3: now
    let a be object;
    assume
A4: a in dom H;
    then
A5: H.a in rng H by FUNCT_1:def 3;
    H.a <> x implies H/(x,y).a = H.a by A4,Def3;
    hence H/(x,y).a = H.a by A1,A2,A5,XBOOLE_0:def 5;
  end;
  dom H = dom (H/(x,y)) by Def3;
  hence thesis by A3,FUNCT_1:2;
end;
