theorem Th70:
  [S,x] is quantifiable implies ((for a holds J,(v.((S_Bound(@
CQCSub_All([S,x],xSQ)))|a)).(NEx_Val(v,S,x,xSQ)+*(x|a)) |= S) iff for a holds J
  ,v.(NEx_Val(v,S,x,xSQ)+*(x|a)) |= S )
proof
  set S1 = CQCSub_All([S,x],xSQ);
  set z = S_Bound(@S1);
  assume
A1: [S,x] is quantifiable;
  thus (for a holds J,(v.(z|a)).(NEx_Val(v,S,x,xSQ)+*(x|a)) |= S) implies for
  a holds J,v.(NEx_Val(v,S,x,xSQ)+*(x|a)) |= S
  proof
    set X = still_not-bound_in S`1;
    assume
A2: for a holds J,(v.(z|a)).(NEx_Val(v,S,x,xSQ)+*(x|a)) |= S;
    let a;
    set V1 = (v.(z|a)).(NEx_Val(v,S,x,xSQ)+*(x|a));
    set V2 = v.(NEx_Val(v,S,x,xSQ)+*(x|a));
    V1|X = V2|X by A1,Th69;
    then
A3: J,V1 |= S`1 iff J,V2 |= S`1 by Th68;
    J,V1 |= S by A2;
    hence thesis by A3;
  end;
  thus (for a holds J,v.(NEx_Val(v,S,x,xSQ)+*(x|a)) |= S) implies for a holds
  J,(v.(z|a)).(NEx_Val(v,S,x,xSQ)+*(x|a)) |= S
  proof
    set X = still_not-bound_in S`1;
    assume
A4: for a holds J,v.(NEx_Val(v,S,x,xSQ)+*(x|a)) |= S;
    let a;
    set V1 = (v.(z|a)).(NEx_Val(v,S,x,xSQ)+*(x|a));
    set V2 = v.(NEx_Val(v,S,x,xSQ)+*(x|a));
    V1|X = V2|X by A1,Th69;
    then
A5: J,V1 |= S`1 iff J,V2 |= S`1 by Th68;
    J,V2 |= S by A4;
    hence thesis by A5;
  end;
end;
