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
  Free H c= variables_in H
proof
  defpred P[ZF-formula] means Free $1 c= variables_in $1;
A1: P[H1] implies P['not' H1]
  proof
    assume Free H1 c= variables_in H1;
    then Free 'not' H1 c= variables_in H1 by Th60;
    hence thesis by Th140;
  end;
A2: P[H1] & P[H2] implies P[H1 '&' H2]
  proof
    assume Free H1 c= variables_in H1 & Free H2 c= variables_in H2;
    then Free H1 \/ Free H2 c= variables_in H1 \/ variables_in H2 by
XBOOLE_1:13;
    then Free H1 '&' H2 c= variables_in H1 \/ variables_in H2 by Th61;
    hence thesis by Th141;
  end;
A3: P[H1] implies P[All(x,H1)]
  proof
    Free H1 \ {x} c= Free H1 by XBOOLE_1:36;
    then
A4: Free All(x,H1) c= Free H1 by Th62;
    variables_in H1 c= variables_in H1 \/ {x } by XBOOLE_1:7;
    then
A5: variables_in H1 c= variables_in All(x, H1) by Th142;
    assume Free H1 c= variables_in H1;
    then Free H1 c= variables_in All(x,H1) by A5;
    hence thesis by A4,XBOOLE_1:1;
  end;
A6: P[x '=' y] & P[x 'in' y]
  proof
    variables_in (x '=' y) = {x,y} & variables_in (x 'in' y) = {x,y} by Th138
,Th139;
    hence thesis by Th58,Th59;
  end;
  for H holds P[H] from ZFInduction(A6,A1,A2,A3);
  hence thesis;
end;
