reserve Al for QC-alphabet;
reserve i,j,k for Nat,
  A,D for non empty set;
reserve f1,f2 for Element of Funcs(Valuations_in(Al,A),BOOLEAN),
  x,x1,y for bound_QC-variable of Al,
  v,v1 for Element of Valuations_in(Al,A);

theorem Th3:
  for p being Element of Funcs(Valuations_in(Al,A),BOOLEAN) holds
FOR_ALL(x,p).v = TRUE iff for v1 st for y st x <> y holds v1.y = v.y holds
p.v1
  = TRUE
proof
  let p be Element of Funcs(Valuations_in(Al,A),BOOLEAN);
A1: now
    assume FOR_ALL(x,p).v = TRUE;
    then ALL{p.v99 where v99 is Element of Valuations_in(Al,A): for y st x <> y
    holds v99.y = v.y} = TRUE by Def2;
    then
A2: not FALSE in {p.v99 where v99 is Element of Valuations_in(Al,A): for y st
    x <> y holds v99.y = v.y} by MARGREL1:17;
    thus for v1 st for y st x <> y holds v1.y = v.y holds p.v1 = TRUE
    proof
      let v1;
      assume for y st x <> y holds v1.y = v.y;
      then not p.v1 = FALSE by A2;
      hence thesis by XBOOLEAN:def 3;
    end;
  end;
  now
    assume for v1 st for y st x <> y holds v1.y = v.y holds p.v1 = TRUE;
    then not ex v1 st p.v1 = FALSE & for y st x <> y holds v1.y = v.y;
    then
    not FALSE in {p.v99 where v99 is Element of Valuations_in(Al,A): for y st
    x <> y holds v99.y = v.y};
    then ALL{p.v99 where v99 is Element of Valuations_in(Al,A): for y st x <> y
    holds v99.y = v.y} = TRUE by MARGREL1:17;
    hence FOR_ALL(x,p).v = TRUE by Def2;
  end;
  hence thesis by A1;
end;
