theorem Th87:
  [S,x] is quantifiable implies for v holds (J,v.NEx_Val(v,S,x,xSQ
) |= All(x,S`1) iff J,v.Val_S(v,CQCSub_All([S,x],xSQ)) |= CQCSub_All([S,x],xSQ)
  )
proof
  set S1 = CQCSub_All([S,x],xSQ);
  assume
A1: [S,x] is quantifiable;
  then S1 = Sub_All([S,x],xSQ) by Def5;
  then S1`1 = All([S,x]`2,([S,x]`1)`1) by A1,Th26;
  then S1`1 = All(x,([S,x]`1)`1);
  then
A2: S1`1 = All(x,S`1);
  let v;
  consider vS1,vS2 such that
A3: ( ( for y st y in dom vS1 holds not y in still_not-bound_in All(x,S
`1))& for y st y in dom vS2 holds vS2.y = v.y )& dom NEx_Val(v,S,x,xSQ) misses
  dom vS2 and
A4: v.Val_S(v,S1) = v.(NEx_Val(v,S,x,xSQ) +* vS1 +* vS2) by A1,Th86;
  J,v.NEx_Val(v,S,x,xSQ) |= All(x,S`1) iff J,v.(NEx_Val(v,S,x,xSQ) +* vS1
  +* vS2) |= All(x,S`1) by A3,Th81;
  hence thesis by A4,A2;
end;
