reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,k,n for Nat,
  p,q for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  f,g for Function,
  P,P9 for QC-pred_symbol of k,Al,
  ll,ll9 for CQC-variable_list of k,Al,
  l1 for FinSequence of QC-variables(Al),
  Sub,Sub9,Sub1 for CQC_Substitution of Al,
  S,S9,S1,S2 for Element of CQC-Sub-WFF(Al),
  s for QC-symbol of Al;
reserve vS,vS1,vS2 for Val_Sub of A,Al;
reserve B for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):],
  SQ for second_Q_comp of B;
reserve B for CQC-WFF-like Element of [:QC-Sub-WFF(Al),
  bound_QC-variables(Al):],
  xSQ for second_Q_comp of [S,x],
  SQ for second_Q_comp of B;
reserve B1 for Element of [:QC-Sub-WFF(Al),bound_QC-variables(Al):];
reserve SQ1 for second_Q_comp of B1;
reserve a for Element of A;

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;
