theorem Th77:
  (for v,vS,vS1,vS2 st (for y st y in dom vS1 holds not y in
  still_not-bound_in p) & (for y st y in dom vS2 holds vS2.y = v.y) & dom vS
misses dom vS2 holds J,v.vS |= p iff J,v.(vS+*vS1+*vS2) |= p) & (for v,vS,vS1,
vS2 st (for y st y in dom vS1 holds not y in still_not-bound_in q) & (for y st
y in dom vS2 holds vS2.y = v.y) & dom vS misses dom vS2 holds J,v.vS |= q iff J
  ,v.(vS+*vS1+*vS2) |= q) implies for v,vS,vS1,vS2 st (for y st y in dom vS1
holds not y in still_not-bound_in p '&' q) & (for y st y in dom vS2 holds vS2.y
  = v.y) & dom vS misses dom vS2 holds J,v.vS |= p '&' q iff J,v.(vS+*vS1+*vS2)
  |= p '&' q
proof
  assume
A1: ( for v,vS,vS1,vS2 st (for y st y in dom vS1 holds not y in
  still_not-bound_in p) & (for y st y in dom vS2 holds vS2.y = v.y) & dom vS
  misses dom vS2 holds J, v.vS |= p iff J,v.(vS+*vS1+*vS2) |= p)& for v,vS,vS1,
vS2 st (for y st y in dom vS1 holds not y in still_not-bound_in q) & (for y st
y in dom vS2 holds vS2.y = v.y) & dom vS misses dom vS2 holds J,v.vS |= q iff J
  ,v.(vS+*vS1+*vS2) |= q;
  let v,vS,vS1,vS2 such that
A2: for y st y in dom vS1 holds not y in still_not-bound_in p '&' q and
A3: ( for y st y in dom vS2 holds vS2.y = v.y)& dom vS misses dom vS2;
A4: for y st y in dom vS1 holds (not y in still_not-bound_in p) & not y in
  still_not-bound_in q
  proof
    let y;
    assume y in dom vS1;
    then not y in still_not-bound_in (p '&' q) by A2;
    then not y in (still_not-bound_in p) \/ (still_not-bound_in q) by
QC_LANG3:10;
    hence thesis by XBOOLE_0:def 3;
  end;
A5: J,v.(vS+*vS1+*vS2) |= p & J,v.(vS+*vS1+*vS2) |= q implies J,v.vS |= p &
  J,v.vS |= q
  proof
    assume
A6: J,v.(vS+*vS1+*vS2) |= p & J,v.(vS+*vS1+*vS2) |= q;
    ( for y st y in dom vS1 holds not y in still_not-bound_in p)& for y
    st y in dom vS1 holds not y in still_not-bound_in q by A4;
    hence thesis by A1,A3,A6;
  end;
  J,v.vS |= p & J,v.vS |= q implies J,v.(vS+*vS1+*vS2) |= p & J,v.(vS+*
  vS1+*vS2) |= q
  proof
    assume
A7: J,v.vS |= p & J,v.vS |= q;
    ( for y st y in dom vS1 holds not y in still_not-bound_in p)& for y
    st y in dom vS1 holds not y in still_not-bound_in q by A4;
    hence thesis by A1,A3,A7;
  end;
  hence thesis by A5,VALUAT_1:18;
end;
