reserve p,p1,p2,q,r,F,G,G1,G2,H,H1,H2 for ZF-formula,
  x,x1,x2,y,y1,y2,z,z1,z2,s,t for Variable,
  a,X for set;
reserve M for non empty set,
  m,m9 for Element of M,
  v,v9 for Function of VAR,M;
reserve i,j for Element of NAT;

theorem Th142:
  variables_in All(x,H) = variables_in H \/ {x}
proof
A1: rng All(x,H) = rng (<*4*>^<*x*>) \/ rng H by FINSEQ_1:31
    .= rng <*4*> \/ rng <*x*> \/ rng H by FINSEQ_1:31
    .= {4} \/ rng <*x*> \/ rng H by FINSEQ_1:39
    .= {4} \/ {x} \/ rng H by FINSEQ_1:39
    .= {4} \/ ({x} \/ rng H) by XBOOLE_1:4;
  thus variables_in All(x,H) c= variables_in H \/ {x}
  proof
    let a be object;
    assume
A2: a in variables_in All(x,H);
    then a <> 4 by Th137;
    then not a in {4} by TARSKI:def 1;
    then a in {x} \/ rng H by A1,A2,XBOOLE_0:def 3;
    then
A3: a in {x} or a in rng H by XBOOLE_0:def 3;
    not a in {0,1,2,3,4} by A2,XBOOLE_0:def 5;
    then a in rng H implies a in variables_in H by XBOOLE_0:def 5;
    hence thesis by A3,XBOOLE_0:def 3;
  end;
  let a be object;
  assume a in variables_in H \/ {x};
  then
A4: a in variables_in H or a in {x} by XBOOLE_0:def 3;
  then a in {x} \/ rng H by XBOOLE_0:def 3;
  then
A5: a in {4} \/ ({x} \/ rng H) by XBOOLE_0:def 3;
  a in rng H & not a in {0,1,2,3,4} or a in {x} & x = a by A4,TARSKI:def 1
,XBOOLE_0:def 5;
  then not a in {0,1,2,3,4} by Th136;
  hence thesis by A1,A5,XBOOLE_0:def 5;
end;
